from sympy.core.function import (Derivative, Function, diff) from sympy.core.mul import Mul from sympy.core.numbers import (I, Rational, pi) from sympy.core.relational import Eq from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.hyperbolic import sinh from sympy.functions.elementary.miscellaneous import sqrt from sympy.matrices.dense import Matrix from sympy.core.containers import Tuple from sympy.functions import exp, cos, sin, log, Ci, Si, erf, erfi from sympy.matrices import dotprodsimp, NonSquareMatrixError from sympy.solvers.ode import dsolve from sympy.solvers.ode.ode import constant_renumber from sympy.solvers.ode.subscheck import checksysodesol from sympy.solvers.ode.systems import (_classify_linear_system, linear_ode_to_matrix, ODEOrderError, ODENonlinearError, _simpsol, _is_commutative_anti_derivative, linodesolve, canonical_odes, dsolve_system, _component_division, _eqs2dict, _dict2graph) from sympy.functions import airyai, airybi from sympy.integrals.integrals import Integral from sympy.simplify.ratsimp import ratsimp from sympy.testing.pytest import raises, slow, tooslow, XFAIL C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C0:11') x = symbols('x') f = Function('f') g = Function('g') h = Function('h') def test_linear_ode_to_matrix(): f, g, h = symbols("f, g, h", cls=Function) t = Symbol("t") funcs = [f(t), g(t), h(t)] f1 = f(t).diff(t) g1 = g(t).diff(t) h1 = h(t).diff(t) f2 = f(t).diff(t, 2) g2 = g(t).diff(t, 2) h2 = h(t).diff(t, 2) eqs_1 = [Eq(f1, g(t)), Eq(g1, f(t))] sol_1 = ([Matrix([[1, 0], [0, 1]]), Matrix([[ 0, 1], [1, 0]])], Matrix([[0],[0]])) assert linear_ode_to_matrix(eqs_1, funcs[:-1], t, 1) == sol_1 eqs_2 = [Eq(f1, f(t) + 2*g(t)), Eq(g1, h(t)), Eq(h1, g(t) + h(t) + f(t))] sol_2 = ([Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]), Matrix([[1, 2, 0], [ 0, 0, 1], [1, 1, 1]])], Matrix([[0], [0], [0]])) assert linear_ode_to_matrix(eqs_2, funcs, t, 1) == sol_2 eqs_3 = [Eq(2*f1 + 3*h1, f(t) + g(t)), Eq(4*h1 + 5*g1, f(t) + h(t)), Eq(5*f1 + 4*g1, g(t) + h(t))] sol_3 = ([Matrix([[2, 0, 3], [0, 5, 4], [5, 4, 0]]), Matrix([[1, 1, 0], [1, 0, 1], [0, 1, 1]])], Matrix([[0], [0], [0]])) assert linear_ode_to_matrix(eqs_3, funcs, t, 1) == sol_3 eqs_4 = [Eq(f2 + h(t), f1 + g(t)), Eq(2*h2 + g2 + g1 + g(t), 0), Eq(3*h1, 4)] sol_4 = ([Matrix([[1, 0, 0], [0, 1, 2], [0, 0, 0]]), Matrix([[1, 0, 0], [0, -1, 0], [0, 0, -3]]), Matrix([[0, 1, -1], [0, -1, 0], [0, 0, 0]])], Matrix([[0], [0], [4]])) assert linear_ode_to_matrix(eqs_4, funcs, t, 2) == sol_4 eqs_5 = [Eq(f2, g(t)), Eq(f1 + g1, f(t))] raises(ODEOrderError, lambda: linear_ode_to_matrix(eqs_5, funcs[:-1], t, 1)) eqs_6 = [Eq(f1, f(t)**2), Eq(g1, f(t) + g(t))] raises(ODENonlinearError, lambda: linear_ode_to_matrix(eqs_6, funcs[:-1], t, 1)) def test__classify_linear_system(): x, y, z, w = symbols('x, y, z, w', cls=Function) t, k, l = symbols('t k l') x1 = diff(x(t), t) y1 = diff(y(t), t) z1 = diff(z(t), t) w1 = diff(w(t), t) x2 = diff(x(t), t, t) y2 = diff(y(t), t, t) funcs = [x(t), y(t)] funcs_2 = funcs + [z(t), w(t)] eqs_1 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * t * x(t) + 3 * y(t) + t)) assert _classify_linear_system(eqs_1, funcs, t) is None eqs_2 = (5 * (x1**2) + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * t * x(t) + 3 * y(t) + t)) sol2 = {'is_implicit': True, 'canon_eqs': [[Eq(Derivative(x(t), t), -sqrt(-12*x(t)/5 + 6*y(t)/5)), Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)], [Eq(Derivative(x(t), t), sqrt(-12*x(t)/5 + 6*y(t)/5)), Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)]]} assert _classify_linear_system(eqs_2, funcs, t) == sol2 eqs_2_1 = [Eq(Derivative(x(t), t), -sqrt(-12*x(t)/5 + 6*y(t)/5)), Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)] assert _classify_linear_system(eqs_2_1, funcs, t) is None eqs_2_2 = [Eq(Derivative(x(t), t), sqrt(-12*x(t)/5 + 6*y(t)/5)), Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)] assert _classify_linear_system(eqs_2_2, funcs, t) is None eqs_3 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * x(t) + 3 * y(t)), (5 * w1 + z(t)), (z1 + w(t))) answer_3 = {'no_of_equation': 4, 'eq': (12*x(t) - 6*y(t) + 5*Derivative(x(t), t), -11*x(t) + 3*y(t) + 2*Derivative(y(t), t), z(t) + 5*Derivative(w(t), t), w(t) + Derivative(z(t), t)), 'func': [x(t), y(t), z(t), w(t)], 'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [Rational(12, 5), Rational(-6, 5), 0, 0], [Rational(-11, 2), Rational(3, 2), 0, 0], [0, 0, 0, 1], [0, 0, Rational(1, 5), 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eqs_3, funcs_2, t) == answer_3 eqs_4 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * x(t) + 3 * y(t)), (z1 - w(t)), (w1 - z(t))) answer_4 = {'no_of_equation': 4, 'eq': (12 * x(t) - 6 * y(t) + 5 * Derivative(x(t), t), -11 * x(t) + 3 * y(t) + 2 * Derivative(y(t), t), -w(t) + Derivative(z(t), t), -z(t) + Derivative(w(t), t)), 'func': [x(t), y(t), z(t), w(t)], 'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [Rational(12, 5), Rational(-6, 5), 0, 0], [Rational(-11, 2), Rational(3, 2), 0, 0], [0, 0, 0, -1], [0, 0, -1, 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eqs_4, funcs_2, t) == answer_4 eqs_5 = (5*x1 + 12*x(t) - 6*(y(t)) + x2, (2*y1 - 11*x(t) + 3*y(t)), (z1 - w(t)), (w1 - z(t))) answer_5 = {'no_of_equation': 4, 'eq': (12*x(t) - 6*y(t) + 5*Derivative(x(t), t) + Derivative(x(t), (t, 2)), -11*x(t) + 3*y(t) + 2*Derivative(y(t), t), -w(t) + Derivative(z(t), t), -z(t) + Derivative(w(t), t)), 'func': [x(t), y(t), z(t), w(t)], 'order': {x(t): 2, y(t): 1, z(t): 1, w(t): 1}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type0', 'is_higher_order': True} assert _classify_linear_system(eqs_5, funcs_2, t) == answer_5 eqs_6 = (Eq(x1, 3*y(t) - 11*z(t)), Eq(y1, 7*z(t) - 3*x(t)), Eq(z1, 11*x(t) - 7*y(t))) answer_6 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)), Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [ 0, -3, 11], [ 3, 0, -7], [-11, 7, 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eqs_6, funcs_2[:-1], t) == answer_6 eqs_7 = (Eq(x1, y(t)), Eq(y1, x(t))) answer_7 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))), 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [ 0, -1], [-1, 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eqs_7, funcs, t) == answer_7 eqs_8 = (Eq(x1, 21*x(t)), Eq(y1, 17*x(t) + 3*y(t)), Eq(z1, 5*x(t) + 7*y(t) + 9*z(t))) answer_8 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 21*x(t)), Eq(Derivative(y(t), t), 17*x(t) + 3*y(t)), Eq(Derivative(z(t), t), 5*x(t) + 7*y(t) + 9*z(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [-21, 0, 0], [-17, -3, 0], [ -5, -7, -9]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eqs_8, funcs_2[:-1], t) == answer_8 eqs_9 = (Eq(x1, 4*x(t) + 5*y(t) + 2*z(t)), Eq(y1, x(t) + 13*y(t) + 9*z(t)), Eq(z1, 32*x(t) + 41*y(t) + 11*z(t))) answer_9 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 4*x(t) + 5*y(t) + 2*z(t)), Eq(Derivative(y(t), t), x(t) + 13*y(t) + 9*z(t)), Eq(Derivative(z(t), t), 32*x(t) + 41*y(t) + 11*z(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [ -4, -5, -2], [ -1, -13, -9], [-32, -41, -11]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eqs_9, funcs_2[:-1], t) == answer_9 eqs_10 = (Eq(3*x1, 4*5*(y(t) - z(t))), Eq(4*y1, 3*5*(z(t) - x(t))), Eq(5*z1, 3*4*(x(t) - y(t)))) answer_10 = {'no_of_equation': 3, 'eq': (Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)), Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)), Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [ 0, Rational(-20, 3), Rational(20, 3)], [Rational(15, 4), 0, Rational(-15, 4)], [Rational(-12, 5), Rational(12, 5), 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eqs_10, funcs_2[:-1], t) == answer_10 eq11 = (Eq(x1, 3*y(t) - 11*z(t)), Eq(y1, 7*z(t) - 3*x(t)), Eq(z1, 11*x(t) - 7*y(t))) sol11 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)), Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [ 0, -3, 11], [ 3, 0, -7], [-11, 7, 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eq11, funcs_2[:-1], t) == sol11 eq12 = (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))) sol12 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))), 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [0, -1], [-1, 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eq12, [x(t), y(t)], t) == sol12 eq13 = (Eq(Derivative(x(t), t), 21*x(t)), Eq(Derivative(y(t), t), 17*x(t) + 3*y(t)), Eq(Derivative(z(t), t), 5*x(t) + 7*y(t) + 9*z(t))) sol13 = {'no_of_equation': 3, 'eq': ( Eq(Derivative(x(t), t), 21 * x(t)), Eq(Derivative(y(t), t), 17 * x(t) + 3 * y(t)), Eq(Derivative(z(t), t), 5 * x(t) + 7 * y(t) + 9 * z(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [-21, 0, 0], [-17, -3, 0], [-5, -7, -9]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eq13, [x(t), y(t), z(t)], t) == sol13 eq14 = ( Eq(Derivative(x(t), t), 4*x(t) + 5*y(t) + 2*z(t)), Eq(Derivative(y(t), t), x(t) + 13*y(t) + 9*z(t)), Eq(Derivative(z(t), t), 32*x(t) + 41*y(t) + 11*z(t))) sol14 = {'no_of_equation': 3, 'eq': ( Eq(Derivative(x(t), t), 4 * x(t) + 5 * y(t) + 2 * z(t)), Eq(Derivative(y(t), t), x(t) + 13 * y(t) + 9 * z(t)), Eq(Derivative(z(t), t), 32 * x(t) + 41 * y(t) + 11 * z(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [-4, -5, -2], [-1, -13, -9], [-32, -41, -11]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eq14, [x(t), y(t), z(t)], t) == sol14 eq15 = (Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)), Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)), Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))) sol15 = {'no_of_equation': 3, 'eq': ( Eq(3 * Derivative(x(t), t), 20 * y(t) - 20 * z(t)), Eq(4 * Derivative(y(t), t), -15 * x(t) + 15 * z(t)), Eq(5 * Derivative(z(t), t), 12 * x(t) - 12 * y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([ [0, Rational(-20, 3), Rational(20, 3)], [Rational(15, 4), 0, Rational(-15, 4)], [Rational(-12, 5), Rational(12, 5), 0]]), 'type_of_equation': 'type1', 'is_general': True} assert _classify_linear_system(eq15, [x(t), y(t), z(t)], t) == sol15 # Constant coefficient homogeneous ODEs eq1 = (Eq(diff(x(t), t), x(t) + y(t) + 9), Eq(diff(y(t), t), 2*x(t) + 5*y(t) + 23)) sol1 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), x(t) + y(t) + 9), Eq(Derivative(y(t), t), 2*x(t) + 5*y(t) + 23)), 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([[-1, -1], [-2, -5]]), 'rhs': Matrix([[ 9], [23]]), 'type_of_equation': 'type2'} assert _classify_linear_system(eq1, funcs, t) == sol1 # Non constant coefficient homogeneous ODEs eq1 = (Eq(diff(x(t), t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t), t), 2*x(t) + 5*t*y(t))) sol1 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), 5*t*x(t) + 2*y(t)), Eq(Derivative(y(t), t), 5*t*y(t) + 2*x(t))), 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': False, 'is_homogeneous': True, 'func_coeff': -Matrix([ [-5*t, -2], [ -2, -5*t]]), 'commutative_antiderivative': Matrix([ [5*t**2/2, 2*t], [ 2*t, 5*t**2/2]]), 'type_of_equation': 'type3', 'is_general': True} assert _classify_linear_system(eq1, funcs, t) == sol1 # Non constant coefficient non-homogeneous ODEs eq1 = [Eq(x1, x(t) + t*y(t) + t), Eq(y1, t*x(t) + y(t))] sol1 = {'no_of_equation': 2, 'eq': [Eq(Derivative(x(t), t), t*y(t) + t + x(t)), Eq(Derivative(y(t), t), t*x(t) + y(t))], 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-1, -t], [-t, -1]]), 'commutative_antiderivative': Matrix([ [ t, t**2/2], [t**2/2, t]]), 'rhs': Matrix([ [t], [0]]), 'type_of_equation': 'type4'} assert _classify_linear_system(eq1, funcs, t) == sol1 eq2 = [Eq(x1, t*x(t) + t*y(t) + t), Eq(y1, t*x(t) + t*y(t) + cos(t))] sol2 = {'no_of_equation': 2, 'eq': [Eq(Derivative(x(t), t), t*x(t) + t*y(t) + t), Eq(Derivative(y(t), t), t*x(t) + t*y(t) + cos(t))], 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ [ t], [cos(t)]]), 'func_coeff': Matrix([ [t, t], [t, t]]), 'is_constant': False, 'type_of_equation': 'type4', 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2], [t**2/2, t**2/2]])} assert _classify_linear_system(eq2, funcs, t) == sol2 eq3 = [Eq(x1, t*(x(t) + y(t) + z(t) + 1)), Eq(y1, t*(x(t) + y(t) + z(t))), Eq(z1, t*(x(t) + y(t) + z(t)))] sol3 = {'no_of_equation': 3, 'eq': [Eq(Derivative(x(t), t), t*(x(t) + y(t) + z(t) + 1)), Eq(Derivative(y(t), t), t*(x(t) + y(t) + z(t))), Eq(Derivative(z(t), t), t*(x(t) + y(t) + z(t)))], 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-t, -t, -t], [-t, -t, -t], [-t, -t, -t]]), 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2]]), 'rhs': Matrix([ [t], [0], [0]]), 'type_of_equation': 'type4'} assert _classify_linear_system(eq3, funcs_2[:-1], t) == sol3 eq4 = [Eq(x1, x(t) + y(t) + t*z(t) + 1), Eq(y1, x(t) + t*y(t) + z(t) + 10), Eq(z1, t*x(t) + y(t) + z(t) + t)] sol4 = {'no_of_equation': 3, 'eq': [Eq(Derivative(x(t), t), t*z(t) + x(t) + y(t) + 1), Eq(Derivative(y(t), t), t*y(t) + x(t) + z(t) + 10), Eq(Derivative(z(t), t), t*x(t) + t + y(t) + z(t))], 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-1, -1, -t], [-1, -t, -1], [-t, -1, -1]]), 'commutative_antiderivative': Matrix([ [ t, t, t**2/2], [ t, t**2/2, t], [t**2/2, t, t]]), 'rhs': Matrix([ [ 1], [10], [ t]]), 'type_of_equation': 'type4'} assert _classify_linear_system(eq4, funcs_2[:-1], t) == sol4 sum_terms = t*(x(t) + y(t) + z(t) + w(t)) eq5 = [Eq(x1, sum_terms), Eq(y1, sum_terms), Eq(z1, sum_terms + 1), Eq(w1, sum_terms)] sol5 = {'no_of_equation': 4, 'eq': [Eq(Derivative(x(t), t), t*(w(t) + x(t) + y(t) + z(t))), Eq(Derivative(y(t), t), t*(w(t) + x(t) + y(t) + z(t))), Eq(Derivative(z(t), t), t*(w(t) + x(t) + y(t) + z(t)) + 1), Eq(Derivative(w(t), t), t*(w(t) + x(t) + y(t) + z(t)))], 'func': [x(t), y(t), z(t), w(t)], 'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1}, 'is_linear': True, 'is_constant': False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-t, -t, -t, -t], [-t, -t, -t, -t], [-t, -t, -t, -t], [-t, -t, -t, -t]]), 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2, t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2, t**2/2]]), 'rhs': Matrix([ [0], [0], [1], [0]]), 'type_of_equation': 'type4'} assert _classify_linear_system(eq5, funcs_2, t) == sol5 # Second Order t_ = symbols("t_") eq1 = (Eq(9*x(t) + 7*y(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) + 3*Derivative(y(t), t), 11*exp(I*t)), Eq(3*x(t) + 12*y(t) + 5*Derivative(x(t), t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)), 2*exp(I*t))) sol1 = {'no_of_equation': 2, 'eq': (Eq(9*x(t) + 7*y(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) + 3*Derivative(y(t), t), 11*exp(I*t)), Eq(3*x(t) + 12*y(t) + 5*Derivative(x(t), t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)), 2*exp(I*t))), 'func': [x(t), y(t)], 'order': {x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ [11*exp(I*t)], [ 2*exp(I*t)]]), 'type_of_equation': 'type0', 'is_second_order': True, 'is_higher_order': True} assert _classify_linear_system(eq1, funcs, t) == sol1 eq2 = (Eq((4*t**2 + 7*t + 1)**2*Derivative(x(t), (t, 2)), 5*x(t) + 35*y(t)), Eq((4*t**2 + 7*t + 1)**2*Derivative(y(t), (t, 2)), x(t) + 9*y(t))) sol2 = {'no_of_equation': 2, 'eq': (Eq((4*t**2 + 7*t + 1)**2*Derivative(x(t), (t, 2)), 5*x(t) + 35*y(t)), Eq((4*t**2 + 7*t + 1)**2*Derivative(y(t), (t, 2)), x(t) + 9*y(t))), 'func': [x(t), y(t)], 'order': {x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type2', 'A0': Matrix([ [Rational(53, 4), 35], [ 1, Rational(69, 4)]]), 'g(t)': sqrt(4*t**2 + 7*t + 1), 'tau': sqrt(33)*log(t - sqrt(33)/8 + Rational(7, 8))/33 - sqrt(33)*log(t + sqrt(33)/8 + Rational(7, 8))/33, 'is_transformed': True, 't_': t_, 'is_second_order': True, 'is_higher_order': True} assert _classify_linear_system(eq2, funcs, t) == sol2 eq3 = ((t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - y(t))*exp(t) + Derivative(x(t), (t, 2)), t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), t) - y(t)) + Derivative(y(t), (t, 2))) sol3 = {'no_of_equation': 2, 'eq': ((t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - y(t))*exp(t) + Derivative(x(t), (t, 2)), t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), t) - y(t)) + Derivative(y(t), (t, 2))), 'func': [x(t), y(t)], 'order': {x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type1', 'A1': Matrix([ [-t*log(t), -t*exp(t)], [ -t**3, -t**2]]), 'is_second_order': True, 'is_higher_order': True} assert _classify_linear_system(eq3, funcs, t) == sol3 eq4 = (Eq(x2, k*x(t) - l*y1), Eq(y2, l*x1 + k*y(t))) sol4 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), (t, 2)), k*x(t) - l*Derivative(y(t), t)), Eq(Derivative(y(t), (t, 2)), k*y(t) + l*Derivative(x(t), t))), 'func': [x(t), y(t)], 'order': {x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type0', 'is_second_order': True, 'is_higher_order': True} assert _classify_linear_system(eq4, funcs, t) == sol4 # Multiple matches f, g = symbols("f g", cls=Function) y, t_ = symbols("y t_") funcs = [f(t), g(t)] eq1 = [Eq(Derivative(f(t), t)**2 - 2*Derivative(f(t), t) + 1, 4), Eq(-y*f(t) + Derivative(g(t), t), 0)] sol1 = {'is_implicit': True, 'canon_eqs': [[Eq(Derivative(f(t), t), -1), Eq(Derivative(g(t), t), y*f(t))], [Eq(Derivative(f(t), t), 3), Eq(Derivative(g(t), t), y*f(t))]]} assert _classify_linear_system(eq1, funcs, t) == sol1 raises(ValueError, lambda: _classify_linear_system(eq1, funcs[:1], t)) eq2 = [Eq(Derivative(f(t), t), (2*f(t) + g(t) + 1)/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)] sol2 = {'no_of_equation': 2, 'eq': [Eq(Derivative(f(t), t), (2*f(t) + g(t) + 1)/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)], 'func': [f(t), g(t)], 'order': {f(t): 1, g(t): 1}, 'is_linear': True, 'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ [1], [0]]), 'func_coeff': Matrix([ [2, 1], [1, 2]]), 'is_constant': False, 'type_of_equation': 'type6', 't_': t_, 'tau': log(t), 'commutative_antiderivative': Matrix([ [2*log(t), log(t)], [ log(t), 2*log(t)]])} assert _classify_linear_system(eq2, funcs, t) == sol2 eq3 = [Eq(Derivative(f(t), t), (2*f(t) + g(t))/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)] sol3 = {'no_of_equation': 2, 'eq': [Eq(Derivative(f(t), t), (2*f(t) + g(t))/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)], 'func': [f(t), g(t)], 'order': {f(t): 1, g(t): 1}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'func_coeff': Matrix([ [2, 1], [1, 2]]), 'is_constant': False, 'type_of_equation': 'type5', 't_': t_, 'rhs': Matrix([ [0], [0]]), 'tau': log(t), 'commutative_antiderivative': Matrix([ [2*log(t), log(t)], [ log(t), 2*log(t)]])} assert _classify_linear_system(eq3, funcs, t) == sol3 def test_matrix_exp(): from sympy.matrices.dense import Matrix, eye, zeros from sympy.solvers.ode.systems import matrix_exp t = Symbol('t') for n in range(1, 6+1): assert matrix_exp(zeros(n), t) == eye(n) for n in range(1, 6+1): A = eye(n) expAt = exp(t) * eye(n) assert matrix_exp(A, t) == expAt for n in range(1, 6+1): A = Matrix(n, n, lambda i,j: i+1 if i==j else 0) expAt = Matrix(n, n, lambda i,j: exp((i+1)*t) if i==j else 0) assert matrix_exp(A, t) == expAt A = Matrix([[0, 1], [-1, 0]]) expAt = Matrix([[cos(t), sin(t)], [-sin(t), cos(t)]]) assert matrix_exp(A, t) == expAt A = Matrix([[2, -5], [2, -4]]) expAt = Matrix([ [3*exp(-t)*sin(t) + exp(-t)*cos(t), -5*exp(-t)*sin(t)], [2*exp(-t)*sin(t), -3*exp(-t)*sin(t) + exp(-t)*cos(t)] ]) assert matrix_exp(A, t) == expAt A = Matrix([[21, 17, 6], [-5, -1, -6], [4, 4, 16]]) # TO update this. # expAt = Matrix([ # [(8*t*exp(12*t) + 5*exp(12*t) - 1)*exp(4*t)/4, # (8*t*exp(12*t) + 5*exp(12*t) - 5)*exp(4*t)/4, # (exp(12*t) - 1)*exp(4*t)/2], # [(-8*t*exp(12*t) - exp(12*t) + 1)*exp(4*t)/4, # (-8*t*exp(12*t) - exp(12*t) + 5)*exp(4*t)/4, # (-exp(12*t) + 1)*exp(4*t)/2], # [4*t*exp(16*t), 4*t*exp(16*t), exp(16*t)]]) expAt = Matrix([ [2*t*exp(16*t) + 5*exp(16*t)/4 - exp(4*t)/4, 2*t*exp(16*t) + 5*exp(16*t)/4 - 5*exp(4*t)/4, exp(16*t)/2 - exp(4*t)/2], [ -2*t*exp(16*t) - exp(16*t)/4 + exp(4*t)/4, -2*t*exp(16*t) - exp(16*t)/4 + 5*exp(4*t)/4, -exp(16*t)/2 + exp(4*t)/2], [ 4*t*exp(16*t), 4*t*exp(16*t), exp(16*t)] ]) assert matrix_exp(A, t) == expAt A = Matrix([[1, 1, 0, 0], [0, 1, 1, 0], [0, 0, 1, -S(1)/8], [0, 0, S(1)/2, S(1)/2]]) expAt = Matrix([ [exp(t), t*exp(t), 4*t*exp(3*t/4) + 8*t*exp(t) + 48*exp(3*t/4) - 48*exp(t), -2*t*exp(3*t/4) - 2*t*exp(t) - 16*exp(3*t/4) + 16*exp(t)], [0, exp(t), -t*exp(3*t/4) - 8*exp(3*t/4) + 8*exp(t), t*exp(3*t/4)/2 + 2*exp(3*t/4) - 2*exp(t)], [0, 0, t*exp(3*t/4)/4 + exp(3*t/4), -t*exp(3*t/4)/8], [0, 0, t*exp(3*t/4)/2, -t*exp(3*t/4)/4 + exp(3*t/4)] ]) assert matrix_exp(A, t) == expAt A = Matrix([ [ 0, 1, 0, 0], [-1, 0, 0, 0], [ 0, 0, 0, 1], [ 0, 0, -1, 0]]) expAt = Matrix([ [ cos(t), sin(t), 0, 0], [-sin(t), cos(t), 0, 0], [ 0, 0, cos(t), sin(t)], [ 0, 0, -sin(t), cos(t)]]) assert matrix_exp(A, t) == expAt A = Matrix([ [ 0, 1, 1, 0], [-1, 0, 0, 1], [ 0, 0, 0, 1], [ 0, 0, -1, 0]]) expAt = Matrix([ [ cos(t), sin(t), t*cos(t), t*sin(t)], [-sin(t), cos(t), -t*sin(t), t*cos(t)], [ 0, 0, cos(t), sin(t)], [ 0, 0, -sin(t), cos(t)]]) assert matrix_exp(A, t) == expAt # This case is unacceptably slow right now but should be solvable... #a, b, c, d, e, f = symbols('a b c d e f') #A = Matrix([ #[-a, b, c, d], #[ a, -b, e, 0], #[ 0, 0, -c - e - f, 0], #[ 0, 0, f, -d]]) A = Matrix([[0, I], [I, 0]]) expAt = Matrix([ [exp(I*t)/2 + exp(-I*t)/2, exp(I*t)/2 - exp(-I*t)/2], [exp(I*t)/2 - exp(-I*t)/2, exp(I*t)/2 + exp(-I*t)/2]]) assert matrix_exp(A, t) == expAt # Testing Errors M = Matrix([[1, 2, 3], [4, 5, 6], [7, 7, 7]]) M1 = Matrix([[t, 1], [1, 1]]) raises(ValueError, lambda: matrix_exp(M[:, :2], t)) raises(ValueError, lambda: matrix_exp(M[:2, :], t)) raises(ValueError, lambda: matrix_exp(M1, t)) raises(ValueError, lambda: matrix_exp(M1[:1, :1], t)) def test_canonical_odes(): f, g, h = symbols('f g h', cls=Function) x = symbols('x') funcs = [f(x), g(x), h(x)] eqs1 = [Eq(f(x).diff(x, x), f(x) + 2*g(x)), Eq(g(x) + 1, g(x).diff(x) + f(x))] sol1 = [[Eq(Derivative(f(x), (x, 2)), f(x) + 2*g(x)), Eq(Derivative(g(x), x), -f(x) + g(x) + 1)]] assert canonical_odes(eqs1, funcs[:2], x) == sol1 eqs2 = [Eq(f(x).diff(x), h(x).diff(x) + f(x)), Eq(g(x).diff(x)**2, f(x) + h(x)), Eq(h(x).diff(x), f(x))] sol2 = [[Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), -sqrt(f(x) + h(x))), Eq(Derivative(h(x), x), f(x))], [Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), sqrt(f(x) + h(x))), Eq(Derivative(h(x), x), f(x))]] assert canonical_odes(eqs2, funcs, x) == sol2 def test_sysode_linear_neq_order1_type1(): f, g, x, y, h = symbols('f g x y h', cls=Function) a, b, c, t = symbols('a b c t') eqs1 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), y(t))] sol1 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) eqs2 = [Eq(Derivative(x(t), t), 2*x(t)), Eq(Derivative(y(t), t), 3*y(t))] sol2 = [Eq(x(t), C1*exp(2*t)), Eq(y(t), C2*exp(3*t))] assert dsolve(eqs2) == sol2 assert checksysodesol(eqs2, sol2) == (True, [0, 0]) eqs3 = [Eq(Derivative(x(t), t), a*x(t)), Eq(Derivative(y(t), t), a*y(t))] sol3 = [Eq(x(t), C1*exp(a*t)), Eq(y(t), C2*exp(a*t))] assert dsolve(eqs3) == sol3 assert checksysodesol(eqs3, sol3) == (True, [0, 0]) # Regression test case for issue #15474 # https://github.com/sympy/sympy/issues/15474 eqs4 = [Eq(Derivative(x(t), t), a*x(t)), Eq(Derivative(y(t), t), b*y(t))] sol4 = [Eq(x(t), C1*exp(a*t)), Eq(y(t), C2*exp(b*t))] assert dsolve(eqs4) == sol4 assert checksysodesol(eqs4, sol4) == (True, [0, 0]) eqs5 = [Eq(Derivative(x(t), t), -y(t)), Eq(Derivative(y(t), t), x(t))] sol5 = [Eq(x(t), -C1*sin(t) - C2*cos(t)), Eq(y(t), C1*cos(t) - C2*sin(t))] assert dsolve(eqs5) == sol5 assert checksysodesol(eqs5, sol5) == (True, [0, 0]) eqs6 = [Eq(Derivative(x(t), t), -2*y(t)), Eq(Derivative(y(t), t), 2*x(t))] sol6 = [Eq(x(t), -C1*sin(2*t) - C2*cos(2*t)), Eq(y(t), C1*cos(2*t) - C2*sin(2*t))] assert dsolve(eqs6) == sol6 assert checksysodesol(eqs6, sol6) == (True, [0, 0]) eqs7 = [Eq(Derivative(x(t), t), I*y(t)), Eq(Derivative(y(t), t), I*x(t))] sol7 = [Eq(x(t), -C1*exp(-I*t) + C2*exp(I*t)), Eq(y(t), C1*exp(-I*t) + C2*exp(I*t))] assert dsolve(eqs7) == sol7 assert checksysodesol(eqs7, sol7) == (True, [0, 0]) eqs8 = [Eq(Derivative(x(t), t), -a*y(t)), Eq(Derivative(y(t), t), a*x(t))] sol8 = [Eq(x(t), -I*C1*exp(-I*a*t) + I*C2*exp(I*a*t)), Eq(y(t), C1*exp(-I*a*t) + C2*exp(I*a*t))] assert dsolve(eqs8) == sol8 assert checksysodesol(eqs8, sol8) == (True, [0, 0]) eqs9 = [Eq(Derivative(x(t), t), x(t) + y(t)), Eq(Derivative(y(t), t), x(t) - y(t))] sol9 = [Eq(x(t), C1*(1 - sqrt(2))*exp(-sqrt(2)*t) + C2*(1 + sqrt(2))*exp(sqrt(2)*t)), Eq(y(t), C1*exp(-sqrt(2)*t) + C2*exp(sqrt(2)*t))] assert dsolve(eqs9) == sol9 assert checksysodesol(eqs9, sol9) == (True, [0, 0]) eqs10 = [Eq(Derivative(x(t), t), x(t) + y(t)), Eq(Derivative(y(t), t), x(t) + y(t))] sol10 = [Eq(x(t), -C1 + C2*exp(2*t)), Eq(y(t), C1 + C2*exp(2*t))] assert dsolve(eqs10) == sol10 assert checksysodesol(eqs10, sol10) == (True, [0, 0]) eqs11 = [Eq(Derivative(x(t), t), 2*x(t) + y(t)), Eq(Derivative(y(t), t), -x(t) + 2*y(t))] sol11 = [Eq(x(t), C1*exp(2*t)*sin(t) + C2*exp(2*t)*cos(t)), Eq(y(t), C1*exp(2*t)*cos(t) - C2*exp(2*t)*sin(t))] assert dsolve(eqs11) == sol11 assert checksysodesol(eqs11, sol11) == (True, [0, 0]) eqs12 = [Eq(Derivative(x(t), t), x(t) + 2*y(t)), Eq(Derivative(y(t), t), 2*x(t) + y(t))] sol12 = [Eq(x(t), -C1*exp(-t) + C2*exp(3*t)), Eq(y(t), C1*exp(-t) + C2*exp(3*t))] assert dsolve(eqs12) == sol12 assert checksysodesol(eqs12, sol12) == (True, [0, 0]) eqs13 = [Eq(Derivative(x(t), t), 4*x(t) + y(t)), Eq(Derivative(y(t), t), -x(t) + 2*y(t))] sol13 = [Eq(x(t), C2*t*exp(3*t) + (C1 + C2)*exp(3*t)), Eq(y(t), -C1*exp(3*t) - C2*t*exp(3*t))] assert dsolve(eqs13) == sol13 assert checksysodesol(eqs13, sol13) == (True, [0, 0]) eqs14 = [Eq(Derivative(x(t), t), a*y(t)), Eq(Derivative(y(t), t), a*x(t))] sol14 = [Eq(x(t), -C1*exp(-a*t) + C2*exp(a*t)), Eq(y(t), C1*exp(-a*t) + C2*exp(a*t))] assert dsolve(eqs14) == sol14 assert checksysodesol(eqs14, sol14) == (True, [0, 0]) eqs15 = [Eq(Derivative(x(t), t), a*y(t)), Eq(Derivative(y(t), t), b*x(t))] sol15 = [Eq(x(t), -C1*a*exp(-t*sqrt(a*b))/sqrt(a*b) + C2*a*exp(t*sqrt(a*b))/sqrt(a*b)), Eq(y(t), C1*exp(-t*sqrt(a*b)) + C2*exp(t*sqrt(a*b)))] assert dsolve(eqs15) == sol15 assert checksysodesol(eqs15, sol15) == (True, [0, 0]) eqs16 = [Eq(Derivative(x(t), t), a*x(t) + b*y(t)), Eq(Derivative(y(t), t), c*x(t))] sol16 = [Eq(x(t), -2*C1*b*exp(t*(a + sqrt(a**2 + 4*b*c))/2)/(a - sqrt(a**2 + 4*b*c)) - 2*C2*b*exp(t*(a - sqrt(a**2 + 4*b*c))/2)/(a + sqrt(a**2 + 4*b*c))), Eq(y(t), C1*exp(t*(a + sqrt(a**2 + 4*b*c))/2) + C2*exp(t*(a - sqrt(a**2 + 4*b*c))/2))] assert dsolve(eqs16) == sol16 assert checksysodesol(eqs16, sol16) == (True, [0, 0]) # Regression test case for issue #18562 # https://github.com/sympy/sympy/issues/18562 eqs17 = [Eq(Derivative(x(t), t), a*y(t) + x(t)), Eq(Derivative(y(t), t), a*x(t) - y(t))] sol17 = [Eq(x(t), C1*a*exp(t*sqrt(a**2 + 1))/(sqrt(a**2 + 1) - 1) - C2*a*exp(-t*sqrt(a**2 + 1))/(sqrt(a**2 + 1) + 1)), Eq(y(t), C1*exp(t*sqrt(a**2 + 1)) + C2*exp(-t*sqrt(a**2 + 1)))] assert dsolve(eqs17) == sol17 assert checksysodesol(eqs17, sol17) == (True, [0, 0]) eqs18 = [Eq(Derivative(x(t), t), 0), Eq(Derivative(y(t), t), 0)] sol18 = [Eq(x(t), C1), Eq(y(t), C2)] assert dsolve(eqs18) == sol18 assert checksysodesol(eqs18, sol18) == (True, [0, 0]) eqs19 = [Eq(Derivative(x(t), t), 2*x(t) - y(t)), Eq(Derivative(y(t), t), x(t))] sol19 = [Eq(x(t), C2*t*exp(t) + (C1 + C2)*exp(t)), Eq(y(t), C1*exp(t) + C2*t*exp(t))] assert dsolve(eqs19) == sol19 assert checksysodesol(eqs19, sol19) == (True, [0, 0]) eqs20 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), x(t) + y(t))] sol20 = [Eq(x(t), C1*exp(t)), Eq(y(t), C1*t*exp(t) + C2*exp(t))] assert dsolve(eqs20) == sol20 assert checksysodesol(eqs20, sol20) == (True, [0, 0]) eqs21 = [Eq(Derivative(x(t), t), 3*x(t)), Eq(Derivative(y(t), t), x(t) + y(t))] sol21 = [Eq(x(t), 2*C1*exp(3*t)), Eq(y(t), C1*exp(3*t) + C2*exp(t))] assert dsolve(eqs21) == sol21 assert checksysodesol(eqs21, sol21) == (True, [0, 0]) eqs22 = [Eq(Derivative(x(t), t), 3*x(t)), Eq(Derivative(y(t), t), y(t))] sol22 = [Eq(x(t), C1*exp(3*t)), Eq(y(t), C2*exp(t))] assert dsolve(eqs22) == sol22 assert checksysodesol(eqs22, sol22) == (True, [0, 0]) @slow def test_sysode_linear_neq_order1_type1_slow(): t = Symbol('t') Z0 = Function('Z0') Z1 = Function('Z1') Z2 = Function('Z2') Z3 = Function('Z3') k01, k10, k20, k21, k23, k30 = symbols('k01 k10 k20 k21 k23 k30') eqs1 = [Eq(Derivative(Z0(t), t), -k01*Z0(t) + k10*Z1(t) + k20*Z2(t) + k30*Z3(t)), Eq(Derivative(Z1(t), t), k01*Z0(t) - k10*Z1(t) + k21*Z2(t)), Eq(Derivative(Z2(t), t), (-k20 - k21 - k23)*Z2(t)), Eq(Derivative(Z3(t), t), k23*Z2(t) - k30*Z3(t))] sol1 = [Eq(Z0(t), C1*k10/k01 - C2*(k10 - k30)*exp(-k30*t)/(k01 + k10 - k30) - C3*(k10*(k20 + k21 - k30) - k20**2 - k20*(k21 + k23 - k30) + k23*k30)*exp(-t*(k20 + k21 + k23))/(k23*(-k01 - k10 + k20 + k21 + k23)) - C4*exp(-t*(k01 + k10))), Eq(Z1(t), C1 - C2*k01*exp(-k30*t)/(k01 + k10 - k30) + C3*(-k01*(k20 + k21 - k30) + k20*k21 + k21**2 + k21*(k23 - k30))*exp(-t*(k20 + k21 + k23))/(k23*(-k01 - k10 + k20 + k21 + k23)) + C4*exp(-t*(k01 + k10))), Eq(Z2(t), -C3*(k20 + k21 + k23 - k30)*exp(-t*(k20 + k21 + k23))/k23), Eq(Z3(t), C2*exp(-k30*t) + C3*exp(-t*(k20 + k21 + k23)))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0, 0, 0]) x, y, z, u, v, w = symbols('x y z u v w', cls=Function) k2, k3 = symbols('k2 k3') a_b, a_c = symbols('a_b a_c', real=True) eqs2 = [Eq(Derivative(z(t), t), k2*y(t)), Eq(Derivative(x(t), t), k3*y(t)), Eq(Derivative(y(t), t), (-k2 - k3)*y(t))] sol2 = [Eq(z(t), C1 - C2*k2*exp(-t*(k2 + k3))/(k2 + k3)), Eq(x(t), -C2*k3*exp(-t*(k2 + k3))/(k2 + k3) + C3), Eq(y(t), C2*exp(-t*(k2 + k3)))] assert dsolve(eqs2) == sol2 assert checksysodesol(eqs2, sol2) == (True, [0, 0, 0]) eqs3 = [4*u(t) - v(t) - 2*w(t) + Derivative(u(t), t), 2*u(t) + v(t) - 2*w(t) + Derivative(v(t), t), 5*u(t) + v(t) - 3*w(t) + Derivative(w(t), t)] sol3 = [Eq(u(t), C3*exp(-2*t) + (C1/2 + sqrt(3)*C2/6)*cos(sqrt(3)*t) + sin(sqrt(3)*t)*(sqrt(3)*C1/6 + C2*Rational(-1, 2))), Eq(v(t), (C1/2 + sqrt(3)*C2/6)*cos(sqrt(3)*t) + sin(sqrt(3)*t)*(sqrt(3)*C1/6 + C2*Rational(-1, 2))), Eq(w(t), C1*cos(sqrt(3)*t) - C2*sin(sqrt(3)*t) + C3*exp(-2*t))] assert dsolve(eqs3) == sol3 assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0]) eqs4 = [Eq(Derivative(x(t), t), w(t)*Rational(-2, 9) + 2*x(t) + y(t) + z(t)*Rational(-8, 9)), Eq(Derivative(y(t), t), w(t)*Rational(4, 9) + 2*y(t) + z(t)*Rational(16, 9)), Eq(Derivative(z(t), t), w(t)*Rational(-2, 9) + z(t)*Rational(37, 9)), Eq(Derivative(w(t), t), w(t)*Rational(44, 9) + z(t)*Rational(-4, 9))] sol4 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t)), Eq(y(t), C2*exp(2*t) + 2*C3*exp(4*t)), Eq(z(t), 2*C3*exp(4*t) + C4*exp(5*t)*Rational(-1, 4)), Eq(w(t), C3*exp(4*t) + C4*exp(5*t))] assert dsolve(eqs4) == sol4 assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0, 0]) # Regression test case for issue #15574 # https://github.com/sympy/sympy/issues/15574 eq5 = [Eq(x(t).diff(t), x(t)), Eq(y(t).diff(t), y(t)), Eq(z(t).diff(t), z(t)), Eq(w(t).diff(t), w(t))] sol5 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t)), Eq(w(t), C4*exp(t))] assert dsolve(eq5) == sol5 assert checksysodesol(eq5, sol5) == (True, [0, 0, 0, 0]) eqs6 = [Eq(Derivative(x(t), t), x(t) + y(t)), Eq(Derivative(y(t), t), y(t) + z(t)), Eq(Derivative(z(t), t), w(t)*Rational(-1, 8) + z(t)), Eq(Derivative(w(t), t), w(t)/2 + z(t)/2)] sol6 = [Eq(x(t), C1*exp(t) + C2*t*exp(t) + 4*C4*t*exp(t*Rational(3, 4)) + (4*C3 + 48*C4)*exp(t*Rational(3, 4))), Eq(y(t), C2*exp(t) - C4*t*exp(t*Rational(3, 4)) - (C3 + 8*C4)*exp(t*Rational(3, 4))), Eq(z(t), C4*t*exp(t*Rational(3, 4))/4 + (C3/4 + C4)*exp(t*Rational(3, 4))), Eq(w(t), C3*exp(t*Rational(3, 4))/2 + C4*t*exp(t*Rational(3, 4))/2)] assert dsolve(eqs6) == sol6 assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0]) # Regression test case for issue #15574 # https://github.com/sympy/sympy/issues/15574 eq7 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), y(t)), Eq(Derivative(z(t), t), z(t)), Eq(Derivative(w(t), t), w(t)), Eq(Derivative(u(t), t), u(t))] sol7 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t)), Eq(w(t), C4*exp(t)), Eq(u(t), C5*exp(t))] assert dsolve(eq7) == sol7 assert checksysodesol(eq7, sol7) == (True, [0, 0, 0, 0, 0]) eqs8 = [Eq(Derivative(x(t), t), 2*x(t) + y(t)), Eq(Derivative(y(t), t), 2*y(t)), Eq(Derivative(z(t), t), 4*z(t)), Eq(Derivative(w(t), t), u(t) + 5*w(t)), Eq(Derivative(u(t), t), 5*u(t))] sol8 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t)), Eq(y(t), C2*exp(2*t)), Eq(z(t), C3*exp(4*t)), Eq(w(t), C4*exp(5*t) + C5*t*exp(5*t)), Eq(u(t), C5*exp(5*t))] assert dsolve(eqs8) == sol8 assert checksysodesol(eqs8, sol8) == (True, [0, 0, 0, 0, 0]) # Regression test case for issue #15574 # https://github.com/sympy/sympy/issues/15574 eq9 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), y(t)), Eq(Derivative(z(t), t), z(t))] sol9 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t))] assert dsolve(eq9) == sol9 assert checksysodesol(eq9, sol9) == (True, [0, 0, 0]) # Regression test case for issue #15407 # https://github.com/sympy/sympy/issues/15407 eqs10 = [Eq(Derivative(x(t), t), (-a_b - a_c)*x(t)), Eq(Derivative(y(t), t), a_b*y(t)), Eq(Derivative(z(t), t), a_c*x(t))] sol10 = [Eq(x(t), -C1*(a_b + a_c)*exp(-t*(a_b + a_c))/a_c), Eq(y(t), C2*exp(a_b*t)), Eq(z(t), C1*exp(-t*(a_b + a_c)) + C3)] assert dsolve(eqs10) == sol10 assert checksysodesol(eqs10, sol10) == (True, [0, 0, 0]) # Regression test case for issue #14312 # https://github.com/sympy/sympy/issues/14312 eqs11 = [Eq(Derivative(x(t), t), k3*y(t)), Eq(Derivative(y(t), t), (-k2 - k3)*y(t)), Eq(Derivative(z(t), t), k2*y(t))] sol11 = [Eq(x(t), C1 + C2*k3*exp(-t*(k2 + k3))/k2), Eq(y(t), -C2*(k2 + k3)*exp(-t*(k2 + k3))/k2), Eq(z(t), C2*exp(-t*(k2 + k3)) + C3)] assert dsolve(eqs11) == sol11 assert checksysodesol(eqs11, sol11) == (True, [0, 0, 0]) # Regression test case for issue #14312 # https://github.com/sympy/sympy/issues/14312 eqs12 = [Eq(Derivative(z(t), t), k2*y(t)), Eq(Derivative(x(t), t), k3*y(t)), Eq(Derivative(y(t), t), (-k2 - k3)*y(t))] sol12 = [Eq(z(t), C1 - C2*k2*exp(-t*(k2 + k3))/(k2 + k3)), Eq(x(t), -C2*k3*exp(-t*(k2 + k3))/(k2 + k3) + C3), Eq(y(t), C2*exp(-t*(k2 + k3)))] assert dsolve(eqs12) == sol12 assert checksysodesol(eqs12, sol12) == (True, [0, 0, 0]) f, g, h = symbols('f, g, h', cls=Function) a, b, c = symbols('a, b, c') # Regression test case for issue #15474 # https://github.com/sympy/sympy/issues/15474 eqs13 = [Eq(Derivative(f(t), t), 2*f(t) + g(t)), Eq(Derivative(g(t), t), a*f(t))] sol13 = [Eq(f(t), C1*exp(t*(sqrt(a + 1) + 1))/(sqrt(a + 1) - 1) - C2*exp(-t*(sqrt(a + 1) - 1))/(sqrt(a + 1) + 1)), Eq(g(t), C1*exp(t*(sqrt(a + 1) + 1)) + C2*exp(-t*(sqrt(a + 1) - 1)))] assert dsolve(eqs13) == sol13 assert checksysodesol(eqs13, sol13) == (True, [0, 0]) eqs14 = [Eq(Derivative(f(t), t), 2*g(t) - 3*h(t)), Eq(Derivative(g(t), t), -2*f(t) + 4*h(t)), Eq(Derivative(h(t), t), 3*f(t) - 4*g(t))] sol14 = [Eq(f(t), 2*C1 - sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(3, 25) + C3*Rational(-8, 25)) - cos(sqrt(29)*t)*(C2*Rational(8, 25) + sqrt(29)*C3*Rational(3, 25))), Eq(g(t), C1*Rational(3, 2) + sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(4, 25) + C3*Rational(6, 25)) - cos(sqrt(29)*t)*(C2*Rational(6, 25) + sqrt(29)*C3*Rational(-4, 25))), Eq(h(t), C1 + C2*cos(sqrt(29)*t) - C3*sin(sqrt(29)*t))] assert dsolve(eqs14) == sol14 assert checksysodesol(eqs14, sol14) == (True, [0, 0, 0]) eqs15 = [Eq(2*Derivative(f(t), t), 12*g(t) - 12*h(t)), Eq(3*Derivative(g(t), t), -8*f(t) + 8*h(t)), Eq(4*Derivative(h(t), t), 6*f(t) - 6*g(t))] sol15 = [Eq(f(t), C1 - sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(6, 13) + C3*Rational(-16, 13)) - cos(sqrt(29)*t)*(C2*Rational(16, 13) + sqrt(29)*C3*Rational(6, 13))), Eq(g(t), C1 + sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(8, 39) + C3*Rational(16, 13)) - cos(sqrt(29)*t)*(C2*Rational(16, 13) + sqrt(29)*C3*Rational(-8, 39))), Eq(h(t), C1 + C2*cos(sqrt(29)*t) - C3*sin(sqrt(29)*t))] assert dsolve(eqs15) == sol15 assert checksysodesol(eqs15, sol15) == (True, [0, 0, 0]) eq16 = (Eq(diff(x(t), t), 21*x(t)), Eq(diff(y(t), t), 17*x(t) + 3*y(t)), Eq(diff(z(t), t), 5*x(t) + 7*y(t) + 9*z(t))) sol16 = [Eq(x(t), 216*C1*exp(21*t)/209), Eq(y(t), 204*C1*exp(21*t)/209 - 6*C2*exp(3*t)/7), Eq(z(t), C1*exp(21*t) + C2*exp(3*t) + C3*exp(9*t))] assert dsolve(eq16) == sol16 assert checksysodesol(eq16, sol16) == (True, [0, 0, 0]) eqs17 = [Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)), Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))] sol17 = [Eq(x(t), C1*Rational(7, 3) - sin(sqrt(179)*t)*(sqrt(179)*C2*Rational(11, 170) + C3*Rational(-21, 170)) - cos(sqrt(179)*t)*(C2*Rational(21, 170) + sqrt(179)*C3*Rational(11, 170))), Eq(y(t), C1*Rational(11, 3) + sin(sqrt(179)*t)*(sqrt(179)*C2*Rational(7, 170) + C3*Rational(33, 170)) - cos(sqrt(179)*t)*(C2*Rational(33, 170) + sqrt(179)*C3*Rational(-7, 170))), Eq(z(t), C1 + C2*cos(sqrt(179)*t) - C3*sin(sqrt(179)*t))] assert dsolve(eqs17) == sol17 assert checksysodesol(eqs17, sol17) == (True, [0, 0, 0]) eqs18 = [Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)), Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)), Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))] sol18 = [Eq(x(t), C1 - sin(5*sqrt(2)*t)*(sqrt(2)*C2*Rational(4, 3) - C3) - cos(5*sqrt(2)*t)*(C2 + sqrt(2)*C3*Rational(4, 3))), Eq(y(t), C1 + sin(5*sqrt(2)*t)*(sqrt(2)*C2*Rational(3, 4) + C3) - cos(5*sqrt(2)*t)*(C2 + sqrt(2)*C3*Rational(-3, 4))), Eq(z(t), C1 + C2*cos(5*sqrt(2)*t) - C3*sin(5*sqrt(2)*t))] assert dsolve(eqs18) == sol18 assert checksysodesol(eqs18, sol18) == (True, [0, 0, 0]) eqs19 = [Eq(Derivative(x(t), t), 4*x(t) - z(t)), Eq(Derivative(y(t), t), 2*x(t) + 2*y(t) - z(t)), Eq(Derivative(z(t), t), 3*x(t) + y(t))] sol19 = [Eq(x(t), C2*t**2*exp(2*t)/2 + t*(2*C2 + C3)*exp(2*t) + (C1 + C2 + 2*C3)*exp(2*t)), Eq(y(t), C2*t**2*exp(2*t)/2 + t*(2*C2 + C3)*exp(2*t) + (C1 + 2*C3)*exp(2*t)), Eq(z(t), C2*t**2*exp(2*t) + t*(3*C2 + 2*C3)*exp(2*t) + (2*C1 + 3*C3)*exp(2*t))] assert dsolve(eqs19) == sol19 assert checksysodesol(eqs19, sol19) == (True, [0, 0, 0]) eqs20 = [Eq(Derivative(x(t), t), 4*x(t) - y(t) - 2*z(t)), Eq(Derivative(y(t), t), 2*x(t) + y(t) - 2*z(t)), Eq(Derivative(z(t), t), 5*x(t) - 3*z(t))] sol20 = [Eq(x(t), C1*exp(2*t) - sin(t)*(C2*Rational(3, 5) + C3/5) - cos(t)*(C2/5 + C3*Rational(-3, 5))), Eq(y(t), -sin(t)*(C2*Rational(3, 5) + C3/5) - cos(t)*(C2/5 + C3*Rational(-3, 5))), Eq(z(t), C1*exp(2*t) - C2*sin(t) + C3*cos(t))] assert dsolve(eqs20) == sol20 assert checksysodesol(eqs20, sol20) == (True, [0, 0, 0]) eq21 = (Eq(diff(x(t), t), 9*y(t)), Eq(diff(y(t), t), 12*x(t))) sol21 = [Eq(x(t), -sqrt(3)*C1*exp(-6*sqrt(3)*t)/2 + sqrt(3)*C2*exp(6*sqrt(3)*t)/2), Eq(y(t), C1*exp(-6*sqrt(3)*t) + C2*exp(6*sqrt(3)*t))] assert dsolve(eq21) == sol21 assert checksysodesol(eq21, sol21) == (True, [0, 0]) eqs22 = [Eq(Derivative(x(t), t), 2*x(t) + 4*y(t)), Eq(Derivative(y(t), t), 12*x(t) + 41*y(t))] sol22 = [Eq(x(t), C1*(39 - sqrt(1713))*exp(t*(sqrt(1713) + 43)/2)*Rational(-1, 24) + C2*(39 + sqrt(1713))*exp(t*(43 - sqrt(1713))/2)*Rational(-1, 24)), Eq(y(t), C1*exp(t*(sqrt(1713) + 43)/2) + C2*exp(t*(43 - sqrt(1713))/2))] assert dsolve(eqs22) == sol22 assert checksysodesol(eqs22, sol22) == (True, [0, 0]) eqs23 = [Eq(Derivative(x(t), t), x(t) + y(t)), Eq(Derivative(y(t), t), -2*x(t) + 2*y(t))] sol23 = [Eq(x(t), (C1/4 + sqrt(7)*C2/4)*cos(sqrt(7)*t/2)*exp(t*Rational(3, 2)) + sin(sqrt(7)*t/2)*(sqrt(7)*C1/4 + C2*Rational(-1, 4))*exp(t*Rational(3, 2))), Eq(y(t), C1*cos(sqrt(7)*t/2)*exp(t*Rational(3, 2)) - C2*sin(sqrt(7)*t/2)*exp(t*Rational(3, 2)))] assert dsolve(eqs23) == sol23 assert checksysodesol(eqs23, sol23) == (True, [0, 0]) # Regression test case for issue #15474 # https://github.com/sympy/sympy/issues/15474 a = Symbol("a", real=True) eq24 = [x(t).diff(t) - a*y(t), y(t).diff(t) + a*x(t)] sol24 = [Eq(x(t), C1*sin(a*t) + C2*cos(a*t)), Eq(y(t), C1*cos(a*t) - C2*sin(a*t))] assert dsolve(eq24) == sol24 assert checksysodesol(eq24, sol24) == (True, [0, 0]) # Regression test case for issue #19150 # https://github.com/sympy/sympy/issues/19150 eqs25 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), (f(t) - 2*g(t) + x(t))/(b*c)), Eq(Derivative(x(t), t), (g(t) - 2*x(t) + y(t))/(b*c)), Eq(Derivative(y(t), t), (h(t) + x(t) - 2*y(t))/(b*c)), Eq(Derivative(h(t), t), 0)] sol25 = [Eq(f(t), -3*C1 + 4*C2), Eq(g(t), -2*C1 + 3*C2 - C3*exp(-2*t/(b*c)) + C4*exp(-t*(sqrt(2) + 2)/(b*c)) + C5*exp(-t*(2 - sqrt(2))/(b*c))), Eq(x(t), -C1 + 2*C2 - sqrt(2)*C4*exp(-t*(sqrt(2) + 2)/(b*c)) + sqrt(2)*C5*exp(-t*(2 - sqrt(2))/(b*c))), Eq(y(t), C2 + C3*exp(-2*t/(b*c)) + C4*exp(-t*(sqrt(2) + 2)/(b*c)) + C5*exp(-t*(2 - sqrt(2))/(b*c))), Eq(h(t), C1)] assert dsolve(eqs25) == sol25 assert checksysodesol(eqs25, sol25) == (True, [0, 0, 0, 0, 0]) eq26 = [Eq(Derivative(f(t), t), 2*f(t)), Eq(Derivative(g(t), t), 3*f(t) + 7*g(t))] sol26 = [Eq(f(t), -5*C1*exp(2*t)/3), Eq(g(t), C1*exp(2*t) + C2*exp(7*t))] assert dsolve(eq26) == sol26 assert checksysodesol(eq26, sol26) == (True, [0, 0]) eq27 = [Eq(Derivative(f(t), t), -9*I*f(t) - 4*g(t)), Eq(Derivative(g(t), t), -4*I*g(t))] sol27 = [Eq(f(t), 4*I*C1*exp(-4*I*t)/5 + C2*exp(-9*I*t)), Eq(g(t), C1*exp(-4*I*t))] assert dsolve(eq27) == sol27 assert checksysodesol(eq27, sol27) == (True, [0, 0]) eq28 = [Eq(Derivative(f(t), t), -9*I*f(t)), Eq(Derivative(g(t), t), -4*I*g(t))] sol28 = [Eq(f(t), C1*exp(-9*I*t)), Eq(g(t), C2*exp(-4*I*t))] assert dsolve(eq28) == sol28 assert checksysodesol(eq28, sol28) == (True, [0, 0]) eq29 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), 0)] sol29 = [Eq(f(t), C1), Eq(g(t), C2)] assert dsolve(eq29) == sol29 assert checksysodesol(eq29, sol29) == (True, [0, 0]) eq30 = [Eq(Derivative(f(t), t), f(t)), Eq(Derivative(g(t), t), 0)] sol30 = [Eq(f(t), C1*exp(t)), Eq(g(t), C2)] assert dsolve(eq30) == sol30 assert checksysodesol(eq30, sol30) == (True, [0, 0]) eq31 = [Eq(Derivative(f(t), t), g(t)), Eq(Derivative(g(t), t), 0)] sol31 = [Eq(f(t), C1 + C2*t), Eq(g(t), C2)] assert dsolve(eq31) == sol31 assert checksysodesol(eq31, sol31) == (True, [0, 0]) eq32 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), f(t))] sol32 = [Eq(f(t), C1), Eq(g(t), C1*t + C2)] assert dsolve(eq32) == sol32 assert checksysodesol(eq32, sol32) == (True, [0, 0]) eq33 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), g(t))] sol33 = [Eq(f(t), C1), Eq(g(t), C2*exp(t))] assert dsolve(eq33) == sol33 assert checksysodesol(eq33, sol33) == (True, [0, 0]) eq34 = [Eq(Derivative(f(t), t), f(t)), Eq(Derivative(g(t), t), I*g(t))] sol34 = [Eq(f(t), C1*exp(t)), Eq(g(t), C2*exp(I*t))] assert dsolve(eq34) == sol34 assert checksysodesol(eq34, sol34) == (True, [0, 0]) eq35 = [Eq(Derivative(f(t), t), I*f(t)), Eq(Derivative(g(t), t), -I*g(t))] sol35 = [Eq(f(t), C1*exp(I*t)), Eq(g(t), C2*exp(-I*t))] assert dsolve(eq35) == sol35 assert checksysodesol(eq35, sol35) == (True, [0, 0]) eq36 = [Eq(Derivative(f(t), t), I*g(t)), Eq(Derivative(g(t), t), 0)] sol36 = [Eq(f(t), I*C1 + I*C2*t), Eq(g(t), C2)] assert dsolve(eq36) == sol36 assert checksysodesol(eq36, sol36) == (True, [0, 0]) eq37 = [Eq(Derivative(f(t), t), I*g(t)), Eq(Derivative(g(t), t), I*f(t))] sol37 = [Eq(f(t), -C1*exp(-I*t) + C2*exp(I*t)), Eq(g(t), C1*exp(-I*t) + C2*exp(I*t))] assert dsolve(eq37) == sol37 assert checksysodesol(eq37, sol37) == (True, [0, 0]) # Multiple systems eq1 = [Eq(Derivative(f(t), t)**2, g(t)**2), Eq(-f(t) + Derivative(g(t), t), 0)] sol1 = [[Eq(f(t), -C1*sin(t) - C2*cos(t)), Eq(g(t), C1*cos(t) - C2*sin(t))], [Eq(f(t), -C1*exp(-t) + C2*exp(t)), Eq(g(t), C1*exp(-t) + C2*exp(t))]] assert dsolve(eq1) == sol1 for sol in sol1: assert checksysodesol(eq1, sol) == (True, [0, 0]) def test_sysode_linear_neq_order1_type2(): f, g, h, k = symbols('f g h k', cls=Function) x, t, a, b, c, d, y = symbols('x t a b c d y') k1, k2 = symbols('k1 k2') eqs1 = [Eq(Derivative(f(x), x), f(x) + g(x) + 5), Eq(Derivative(g(x), x), -f(x) - g(x) + 7)] sol1 = [Eq(f(x), C1 + C2 + 6*x**2 + x*(C2 + 5)), Eq(g(x), -C1 - 6*x**2 - x*(C2 - 7))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) eqs2 = [Eq(Derivative(f(x), x), f(x) + g(x) + 5), Eq(Derivative(g(x), x), f(x) + g(x) + 7)] sol2 = [Eq(f(x), -C1 + C2*exp(2*x) - x - 3), Eq(g(x), C1 + C2*exp(2*x) + x - 3)] assert dsolve(eqs2) == sol2 assert checksysodesol(eqs2, sol2) == (True, [0, 0]) eqs3 = [Eq(Derivative(f(x), x), f(x) + 5), Eq(Derivative(g(x), x), f(x) + 7)] sol3 = [Eq(f(x), C1*exp(x) - 5), Eq(g(x), C1*exp(x) + C2 + 2*x - 5)] assert dsolve(eqs3) == sol3 assert checksysodesol(eqs3, sol3) == (True, [0, 0]) eqs4 = [Eq(Derivative(f(x), x), f(x) + exp(x)), Eq(Derivative(g(x), x), x*exp(x) + f(x) + g(x))] sol4 = [Eq(f(x), C1*exp(x) + x*exp(x)), Eq(g(x), C1*x*exp(x) + C2*exp(x) + x**2*exp(x))] assert dsolve(eqs4) == sol4 assert checksysodesol(eqs4, sol4) == (True, [0, 0]) eqs5 = [Eq(Derivative(f(x), x), 5*x + f(x) + g(x)), Eq(Derivative(g(x), x), f(x) - g(x))] sol5 = [Eq(f(x), C1*(1 + sqrt(2))*exp(sqrt(2)*x) + C2*(1 - sqrt(2))*exp(-sqrt(2)*x) + x*Rational(-5, 2) + Rational(-5, 2)), Eq(g(x), C1*exp(sqrt(2)*x) + C2*exp(-sqrt(2)*x) + x*Rational(-5, 2))] assert dsolve(eqs5) == sol5 assert checksysodesol(eqs5, sol5) == (True, [0, 0]) eqs6 = [Eq(Derivative(f(x), x), -9*f(x) - 4*g(x)), Eq(Derivative(g(x), x), -4*g(x)), Eq(Derivative(h(x), x), h(x) + exp(x))] sol6 = [Eq(f(x), C2*exp(-4*x)*Rational(-4, 5) + C1*exp(-9*x)), Eq(g(x), C2*exp(-4*x)), Eq(h(x), C3*exp(x) + x*exp(x))] assert dsolve(eqs6) == sol6 assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0]) # Regression test case for issue #8859 # https://github.com/sympy/sympy/issues/8859 eqs7 = [Eq(Derivative(f(t), t), 3*t + f(t)), Eq(Derivative(g(t), t), g(t))] sol7 = [Eq(f(t), C1*exp(t) - 3*t - 3), Eq(g(t), C2*exp(t))] assert dsolve(eqs7) == sol7 assert checksysodesol(eqs7, sol7) == (True, [0, 0]) # Regression test case for issue #8567 # https://github.com/sympy/sympy/issues/8567 eqs8 = [Eq(Derivative(f(t), t), f(t) + 2*g(t)), Eq(Derivative(g(t), t), -2*f(t) + g(t) + 2*exp(t))] sol8 = [Eq(f(t), C1*exp(t)*sin(2*t) + C2*exp(t)*cos(2*t) + exp(t)*sin(2*t)**2 + exp(t)*cos(2*t)**2), Eq(g(t), C1*exp(t)*cos(2*t) - C2*exp(t)*sin(2*t))] assert dsolve(eqs8) == sol8 assert checksysodesol(eqs8, sol8) == (True, [0, 0]) # Regression test case for issue #19150 # https://github.com/sympy/sympy/issues/19150 eqs9 = [Eq(Derivative(f(t), t), (c - 2*f(t) + g(t))/(a*b)), Eq(Derivative(g(t), t), (f(t) - 2*g(t) + h(t))/(a*b)), Eq(Derivative(h(t), t), (d + g(t) - 2*h(t))/(a*b))] sol9 = [Eq(f(t), -C1*exp(-2*t/(a*b)) + C2*exp(-t*(sqrt(2) + 2)/(a*b)) + C3*exp(-t*(2 - sqrt(2))/(a*b)) + Mul(Rational(1, 4), 3*c + d, evaluate=False)), Eq(g(t), -sqrt(2)*C2*exp(-t*(sqrt(2) + 2)/(a*b)) + sqrt(2)*C3*exp(-t*(2 - sqrt(2))/(a*b)) + Mul(Rational(1, 2), c + d, evaluate=False)), Eq(h(t), C1*exp(-2*t/(a*b)) + C2*exp(-t*(sqrt(2) + 2)/(a*b)) + C3*exp(-t*(2 - sqrt(2))/(a*b)) + Mul(Rational(1, 4), c + 3*d, evaluate=False))] assert dsolve(eqs9) == sol9 assert checksysodesol(eqs9, sol9) == (True, [0, 0, 0]) # Regression test case for issue #16635 # https://github.com/sympy/sympy/issues/16635 eqs10 = [Eq(Derivative(f(t), t), 15*t + f(t) - g(t) - 10), Eq(Derivative(g(t), t), -15*t + f(t) - g(t) - 5)] sol10 = [Eq(f(t), C1 + C2 + 5*t**3 + 5*t**2 + t*(C2 - 10)), Eq(g(t), C1 + 5*t**3 - 10*t**2 + t*(C2 - 5))] assert dsolve(eqs10) == sol10 assert checksysodesol(eqs10, sol10) == (True, [0, 0]) # Multiple solutions eqs11 = [Eq(Derivative(f(t), t)**2 - 2*Derivative(f(t), t) + 1, 4), Eq(-y*f(t) + Derivative(g(t), t), 0)] sol11 = [[Eq(f(t), C1 - t), Eq(g(t), C1*t*y + C2*y + t**2*y*Rational(-1, 2))], [Eq(f(t), C1 + 3*t), Eq(g(t), C1*t*y + C2*y + t**2*y*Rational(3, 2))]] assert dsolve(eqs11) == sol11 for s11 in sol11: assert checksysodesol(eqs11, s11) == (True, [0, 0]) # test case for issue #19831 # https://github.com/sympy/sympy/issues/19831 n = symbols('n', positive=True) x0 = symbols('x_0') t0 = symbols('t_0') x_0 = symbols('x_0') t_0 = symbols('t_0') t = symbols('t') x = Function('x') y = Function('y') T = symbols('T') eqs12 = [Eq(Derivative(y(t), t), x(t)), Eq(Derivative(x(t), t), n*(y(t) + 1))] sol12 = [Eq(y(t), C1*exp(sqrt(n)*t)*n**Rational(-1, 2) - C2*exp(-sqrt(n)*t)*n**Rational(-1, 2) - 1), Eq(x(t), C1*exp(sqrt(n)*t) + C2*exp(-sqrt(n)*t))] assert dsolve(eqs12) == sol12 assert checksysodesol(eqs12, sol12) == (True, [0, 0]) sol12b = [ Eq(y(t), (T*exp(-sqrt(n)*t_0)/2 + exp(-sqrt(n)*t_0)/2 + x_0*exp(-sqrt(n)*t_0)/(2*sqrt(n)))*exp(sqrt(n)*t) + (T*exp(sqrt(n)*t_0)/2 + exp(sqrt(n)*t_0)/2 - x_0*exp(sqrt(n)*t_0)/(2*sqrt(n)))*exp(-sqrt(n)*t) - 1), Eq(x(t), (T*sqrt(n)*exp(-sqrt(n)*t_0)/2 + sqrt(n)*exp(-sqrt(n)*t_0)/2 + x_0*exp(-sqrt(n)*t_0)/2)*exp(sqrt(n)*t) - (T*sqrt(n)*exp(sqrt(n)*t_0)/2 + sqrt(n)*exp(sqrt(n)*t_0)/2 - x_0*exp(sqrt(n)*t_0)/2)*exp(-sqrt(n)*t)) ] assert dsolve(eqs12, ics={y(t0): T, x(t0): x0}) == sol12b assert checksysodesol(eqs12, sol12b) == (True, [0, 0]) #Test cases added for the issue 19763 #https://github.com/sympy/sympy/issues/19763 eq13 = [Eq(Derivative(f(t), t), f(t) + g(t) + 9), Eq(Derivative(g(t), t), 2*f(t) + 5*g(t) + 23)] sol13 = [Eq(f(t), -C1*(2 + sqrt(6))*exp(t*(3 - sqrt(6)))/2 - C2*(2 - sqrt(6))*exp(t*(sqrt(6) + 3))/2 - Rational(22,3)), Eq(g(t), C1*exp(t*(3 - sqrt(6))) + C2*exp(t*(sqrt(6) + 3)) - Rational(5,3))] assert dsolve(eq13) == sol13 assert checksysodesol(eq13, sol13) == (True, [0, 0]) eq14 = [Eq(Derivative(f(t), t), f(t) + g(t) + 81), Eq(Derivative(g(t), t), -2*f(t) + g(t) + 23)] sol14 = [Eq(f(t), sqrt(2)*C1*exp(t)*sin(sqrt(2)*t)/2 + sqrt(2)*C2*exp(t)*cos(sqrt(2)*t)/2 - 58*sin(sqrt(2)*t)**2/3 - 58*cos(sqrt(2)*t)**2/3), Eq(g(t), C1*exp(t)*cos(sqrt(2)*t) - C2*exp(t)*sin(sqrt(2)*t) - 185*sin(sqrt(2)*t)**2/3 - 185*cos(sqrt(2)*t)**2/3)] assert dsolve(eq14) == sol14 assert checksysodesol(eq14, sol14) == (True, [0,0]) eq15 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1), Eq(Derivative(g(t), t), 3*f(t) + 4*g(t) + k2)] sol15 = [Eq(f(t), -C1*(3 - sqrt(33))*exp(t*(5 + sqrt(33))/2)/6 - C2*(3 + sqrt(33))*exp(t*(5 - sqrt(33))/2)/6 + 2*k1 - k2), Eq(g(t), C1*exp(t*(5 + sqrt(33))/2) + C2*exp(t*(5 - sqrt(33))/2) - Mul(Rational(1,2), 3*k1 - k2, evaluate = False))] assert dsolve(eq15) == sol15 assert checksysodesol(eq15, sol15) == (True, [0,0]) eq16 = [Eq(Derivative(f(t), t), k1), Eq(Derivative(g(t), t), k2)] sol16 = [Eq(f(t), C1 + k1*t), Eq(g(t), C2 + k2*t)] assert dsolve(eq16) == sol16 assert checksysodesol(eq16, sol16) == (True, [0,0]) eq17 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), c*f(t) + k2)] sol17 = [Eq(f(t), C1), Eq(g(t), C2*c + t*(C1*c + k2))] assert dsolve(eq17) == sol17 assert checksysodesol(eq17 , sol17) == (True , [0,0]) eq18 = [Eq(Derivative(f(t), t), k1), Eq(Derivative(g(t), t), f(t) + k2)] sol18 = [Eq(f(t), C1 + k1*t), Eq(g(t), C2 + k1*t**2/2 + t*(C1 + k2))] assert dsolve(eq18) == sol18 assert checksysodesol(eq18 , sol18) == (True , [0,0]) eq19 = [Eq(Derivative(f(t), t), k1), Eq(Derivative(g(t), t), f(t) + 2*g(t) + k2)] sol19 = [Eq(f(t), -2*C1 + k1*t), Eq(g(t), C1 + C2*exp(2*t) - k1*t/2 - Mul(Rational(1,4), k1 + 2*k2 , evaluate = False))] assert dsolve(eq19) == sol19 assert checksysodesol(eq19 , sol19) == (True , [0,0]) eq20 = [Eq(diff(f(t), t), f(t) + k1), Eq(diff(g(t), t), k2)] sol20 = [Eq(f(t), C1*exp(t) - k1), Eq(g(t), C2 + k2*t)] assert dsolve(eq20) == sol20 assert checksysodesol(eq20 , sol20) == (True , [0,0]) eq21 = [Eq(diff(f(t), t), g(t) + k1), Eq(diff(g(t), t), 0)] sol21 = [Eq(f(t), C1 + t*(C2 + k1)), Eq(g(t), C2)] assert dsolve(eq21) == sol21 assert checksysodesol(eq21 , sol21) == (True , [0,0]) eq22 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1), Eq(Derivative(g(t), t), k2)] sol22 = [Eq(f(t), -2*C1 + C2*exp(t) - k1 - 2*k2*t - 2*k2), Eq(g(t), C1 + k2*t)] assert dsolve(eq22) == sol22 assert checksysodesol(eq22 , sol22) == (True , [0,0]) eq23 = [Eq(Derivative(f(t), t), g(t) + k1), Eq(Derivative(g(t), t), 2*g(t) + k2)] sol23 = [Eq(f(t), C1 + C2*exp(2*t)/2 - k2/4 + t*(2*k1 - k2)/2), Eq(g(t), C2*exp(2*t) - k2/2)] assert dsolve(eq23) == sol23 assert checksysodesol(eq23 , sol23) == (True , [0,0]) eq24 = [Eq(Derivative(f(t), t), f(t) + k1), Eq(Derivative(g(t), t), 2*f(t) + k2)] sol24 = [Eq(f(t), C1*exp(t)/2 - k1), Eq(g(t), C1*exp(t) + C2 - 2*k1 - t*(2*k1 - k2))] assert dsolve(eq24) == sol24 assert checksysodesol(eq24 , sol24) == (True , [0,0]) eq25 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1), Eq(Derivative(g(t), t), 3*f(t) + 6*g(t) + k2)] sol25 = [Eq(f(t), -2*C1 + C2*exp(7*t)/3 + 2*t*(3*k1 - k2)/7 - Mul(Rational(1,49), k1 + 2*k2 , evaluate = False)), Eq(g(t), C1 + C2*exp(7*t) - t*(3*k1 - k2)/7 - Mul(Rational(3,49), k1 + 2*k2 , evaluate = False))] assert dsolve(eq25) == sol25 assert checksysodesol(eq25 , sol25) == (True , [0,0]) eq26 = [Eq(Derivative(f(t), t), 2*f(t) - g(t) + k1), Eq(Derivative(g(t), t), 4*f(t) - 2*g(t) + 2*k1)] sol26 = [Eq(f(t), C1 + 2*C2 + t*(2*C1 + k1)), Eq(g(t), 4*C2 + t*(4*C1 + 2*k1))] assert dsolve(eq26) == sol26 assert checksysodesol(eq26 , sol26) == (True , [0,0]) # Test Case added for issue #22715 # https://github.com/sympy/sympy/issues/22715 eq27 = [Eq(diff(x(t),t),-1*y(t)+10), Eq(diff(y(t),t),5*x(t)-2*y(t)+3)] sol27 = [Eq(x(t), (C1/5 - 2*C2/5)*exp(-t)*cos(2*t) - (2*C1/5 + C2/5)*exp(-t)*sin(2*t) + 17*sin(2*t)**2/5 + 17*cos(2*t)**2/5), Eq(y(t), C1*exp(-t)*cos(2*t) - C2*exp(-t)*sin(2*t) + 10*sin(2*t)**2 + 10*cos(2*t)**2)] assert dsolve(eq27) == sol27 assert checksysodesol(eq27 , sol27) == (True , [0,0]) def test_sysode_linear_neq_order1_type3(): f, g, h, k, x0 , y0 = symbols('f g h k x0 y0', cls=Function) x, t, a = symbols('x t a') r = symbols('r', real=True) eqs1 = [Eq(Derivative(f(r), r), r*g(r) + f(r)), Eq(Derivative(g(r), r), -r*f(r) + g(r))] sol1 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2)), Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) eqs2 = [Eq(Derivative(f(x), x), x**2*g(x) + x*f(x)), Eq(Derivative(g(x), x), 2*x**2*f(x) + (3*x**2 + x)*g(x))] sol2 = [Eq(f(x), (sqrt(17)*C1/17 + C2*(17 - 3*sqrt(17))/34)*exp(x**3*(3 + sqrt(17))/6 + x**2/2) - exp(x**3*(3 - sqrt(17))/6 + x**2/2)*(sqrt(17)*C1/17 + C2*(3*sqrt(17) + 17)*Rational(-1, 34))), Eq(g(x), exp(x**3*(3 - sqrt(17))/6 + x**2/2)*(C1*(17 - 3*sqrt(17))/34 + sqrt(17)*C2*Rational(-2, 17)) + exp(x**3*(3 + sqrt(17))/6 + x**2/2)*(C1*(3*sqrt(17) + 17)/34 + sqrt(17)*C2*Rational(2, 17)))] assert dsolve(eqs2) == sol2 assert checksysodesol(eqs2, sol2) == (True, [0, 0]) eqs3 = [Eq(f(x).diff(x), x*f(x) + g(x)), Eq(g(x).diff(x), -f(x) + x*g(x))] sol3 = [Eq(f(x), (C1/2 + I*C2/2)*exp(x**2/2 - I*x) + exp(x**2/2 + I*x)*(C1/2 + I*C2*Rational(-1, 2))), Eq(g(x), (I*C1/2 + C2/2)*exp(x**2/2 + I*x) - exp(x**2/2 - I*x)*(I*C1/2 + C2*Rational(-1, 2)))] assert dsolve(eqs3) == sol3 assert checksysodesol(eqs3, sol3) == (True, [0, 0]) eqs4 = [Eq(f(x).diff(x), x*(f(x) + g(x) + h(x))), Eq(g(x).diff(x), x*(f(x) + g(x) + h(x))), Eq(h(x).diff(x), x*(f(x) + g(x) + h(x)))] sol4 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)), Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)), Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2))] assert dsolve(eqs4) == sol4 assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0]) eqs5 = [Eq(f(x).diff(x), x**2*(f(x) + g(x) + h(x))), Eq(g(x).diff(x), x**2*(f(x) + g(x) + h(x))), Eq(h(x).diff(x), x**2*(f(x) + g(x) + h(x)))] sol5 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3)), Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3)), Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3))] assert dsolve(eqs5) == sol5 assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0]) eqs6 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x) + k(x))), Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x) + k(x))), Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x) + k(x))), Eq(Derivative(k(x), x), x*(f(x) + g(x) + h(x) + k(x)))] sol6 = [Eq(f(x), -C1/4 - C2/4 - C3/4 + 3*C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)), Eq(g(x), 3*C1/4 - C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)), Eq(h(x), -C1/4 + 3*C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)), Eq(k(x), -C1/4 - C2/4 + 3*C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2))] assert dsolve(eqs6) == sol6 assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0]) y = symbols("y", real=True) eqs7 = [Eq(Derivative(f(y), y), y*f(y) + g(y)), Eq(Derivative(g(y), y), y*g(y) - f(y))] sol7 = [Eq(f(y), C1*exp(y**2/2)*sin(y) + C2*exp(y**2/2)*cos(y)), Eq(g(y), C1*exp(y**2/2)*cos(y) - C2*exp(y**2/2)*sin(y))] assert dsolve(eqs7) == sol7 assert checksysodesol(eqs7, sol7) == (True, [0, 0]) #Test cases added for the issue 19763 #https://github.com/sympy/sympy/issues/19763 eqs8 = [Eq(Derivative(f(t), t), 5*t*f(t) + 2*h(t)), Eq(Derivative(h(t), t), 2*f(t) + 5*t*h(t))] sol8 = [Eq(f(t), Mul(-1, (C1/2 - C2/2), evaluate = False)*exp(5*t**2/2 - 2*t) + (C1/2 + C2/2)*exp(5*t**2/2 + 2*t)), Eq(h(t), (C1/2 - C2/2)*exp(5*t**2/2 - 2*t) + (C1/2 + C2/2)*exp(5*t**2/2 + 2*t))] assert dsolve(eqs8) == sol8 assert checksysodesol(eqs8, sol8) == (True, [0, 0]) eqs9 = [Eq(diff(f(t), t), 5*t*f(t) + t**2*g(t)), Eq(diff(g(t), t), -t**2*f(t) + 5*t*g(t))] sol9 = [Eq(f(t), (C1/2 - I*C2/2)*exp(I*t**3/3 + 5*t**2/2) + (C1/2 + I*C2/2)*exp(-I*t**3/3 + 5*t**2/2)), Eq(g(t), Mul(-1, (I*C1/2 - C2/2) , evaluate = False)*exp(-I*t**3/3 + 5*t**2/2) + (I*C1/2 + C2/2)*exp(I*t**3/3 + 5*t**2/2))] assert dsolve(eqs9) == sol9 assert checksysodesol(eqs9 , sol9) == (True , [0,0]) eqs10 = [Eq(diff(f(t), t), t**2*g(t) + 5*t*f(t)), Eq(diff(g(t), t), -t**2*f(t) + (9*t**2 + 5*t)*g(t))] sol10 = [Eq(f(t), (C1*(77 - 9*sqrt(77))/154 + sqrt(77)*C2/77)*exp(t**3*(sqrt(77) + 9)/6 + 5*t**2/2) + (C1*(77 + 9*sqrt(77))/154 - sqrt(77)*C2/77)*exp(t**3*(9 - sqrt(77))/6 + 5*t**2/2)), Eq(g(t), (sqrt(77)*C1/77 + C2*(77 - 9*sqrt(77))/154)*exp(t**3*(9 - sqrt(77))/6 + 5*t**2/2) - (sqrt(77)*C1/77 - C2*(77 + 9*sqrt(77))/154)*exp(t**3*(sqrt(77) + 9)/6 + 5*t**2/2))] assert dsolve(eqs10) == sol10 assert checksysodesol(eqs10 , sol10) == (True , [0,0]) eqs11 = [Eq(diff(f(t), t), 5*t*f(t) + t**2*g(t)), Eq(diff(g(t), t), (1-t**2)*f(t) + (5*t + 9*t**2)*g(t))] sol11 = [Eq(f(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), Eq(g(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))] assert dsolve(eqs11) == sol11 @slow def test_sysode_linear_neq_order1_type4(): f, g, h, k = symbols('f g h k', cls=Function) x, t, a = symbols('x t a') r = symbols('r', real=True) eqs1 = [Eq(diff(f(r), r), f(r) + r*g(r) + r**2), Eq(diff(g(r), r), -r*f(r) + g(r) + r)] sol1 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2) + exp(r)*sin(r**2/2)*Integral(r**2*exp(-r)*sin(r**2/2) + r*exp(-r)*cos(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r**2*exp(-r)*cos(r**2/2) - r*exp(-r)*sin(r**2/2), r)), Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2) - exp(r)*sin(r**2/2)*Integral(r**2*exp(-r)*cos(r**2/2) - r*exp(-r)*sin(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r**2*exp(-r)*sin(r**2/2) + r*exp(-r)*cos(r**2/2), r))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) eqs2 = [Eq(diff(f(r), r), f(r) + r*g(r) + r), Eq(diff(g(r), r), -r*f(r) + g(r) + log(r))] sol2 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2) + exp(r)*sin(r**2/2)*Integral(r*exp(-r)*sin(r**2/2) + exp(-r)*log(r)*cos(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r*exp(-r)*cos(r**2/2) - exp(-r)*log(r)*sin( r**2/2), r)), Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2) - exp(r)*sin(r**2/2)*Integral(r*exp(-r)*cos(r**2/2) - exp(-r)*log(r)*sin(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r*exp(-r)*sin(r**2/2) + exp(-r)*log(r)*cos( r**2/2), r))] # XXX: dsolve hangs for this in integration assert dsolve_system(eqs2, simplify=False, doit=False) == [sol2] assert checksysodesol(eqs2, sol2) == (True, [0, 0]) eqs3 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x)) + x), Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x)) + x), Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x)) + 1)] sol3 = [Eq(f(x), C1*Rational(-1, 3) + C2*Rational(-1, 3) + C3*Rational(2, 3) + x**2/6 + x*Rational(-1, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9)), Eq(g(x), C1*Rational(2, 3) + C2*Rational(-1, 3) + C3*Rational(-1, 3) + x**2/6 + x*Rational(-1, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9)), Eq(h(x), C1*Rational(-1, 3) + C2*Rational(2, 3) + C3*Rational(-1, 3) + x**2*Rational(-1, 3) + x*Rational(2, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9))] assert dsolve(eqs3) == sol3 assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0]) eqs4 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x)) + sin(x)), Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x)) + sin(x)), Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x)) + sin(x))] sol4 = [Eq(f(x), C1*Rational(-1, 3) + C2*Rational(-1, 3) + C3*Rational(2, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3, 2))), Eq(g(x), C1*Rational(2, 3) + C2*Rational(-1, 3) + C3*Rational(-1, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3, 2))), Eq(h(x), C1*Rational(-1, 3) + C2*Rational(2, 3) + C3*Rational(-1, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3, 2)))] assert dsolve(eqs4) == sol4 assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0]) eqs5 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)), Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)), Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)), Eq(Derivative(k(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1))] sol5 = [Eq(f(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(3, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)), Eq(g(x), C1*Rational(3, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)), Eq(h(x), C1*Rational(-1, 4) + C2*Rational(3, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)), Eq(k(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(3, 4) + C4*Rational(-1, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4))] assert dsolve(eqs5) == sol5 assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0, 0]) eqs6 = [Eq(Derivative(f(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)), Eq(Derivative(g(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)), Eq(Derivative(h(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)), Eq(Derivative(k(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1))] sol6 = [Eq(f(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(3, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)), Eq(g(x), C1*Rational(3, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)), Eq(h(x), C1*Rational(-1, 4) + C2*Rational(3, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)), Eq(k(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(3, 4) + C4*Rational(-1, 4) + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4))] assert dsolve(eqs6) == sol6 assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0]) eqs7 = [Eq(Derivative(f(x), x), (f(x) + g(x) + h(x))*log(x) + sin(x)), Eq(Derivative(g(x), x), (f(x) + g(x) + h(x))*log(x) + sin(x)), Eq(Derivative(h(x), x), (f(x) + g(x) + h(x))*log(x) + sin(x))] sol7 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) - 3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x)), Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) - 3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x)), Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) - 3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x))] with dotprodsimp(True): assert dsolve(eqs7, simplify=False, doit=False) == sol7 assert checksysodesol(eqs7, sol7) == (True, [0, 0, 0]) eqs8 = [Eq(Derivative(f(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x)), Eq(Derivative(g(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x)), Eq(Derivative(h(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x)), Eq(Derivative(k(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x))] sol8 = [Eq(f(x), -C1/4 - C2/4 - C3/4 + 3*C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)), Eq(g(x), 3*C1/4 - C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)), Eq(h(x), -C1/4 + 3*C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)), Eq(k(x), -C1/4 - C2/4 + 3*C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) - 4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x))] with dotprodsimp(True): assert dsolve(eqs8) == sol8 assert checksysodesol(eqs8, sol8) == (True, [0, 0, 0, 0]) def test_sysode_linear_neq_order1_type5_type6(): f, g = symbols("f g", cls=Function) x, x_ = symbols("x x_") # Type 5 eqs1 = [Eq(Derivative(f(x), x), (2*f(x) + g(x))/x), Eq(Derivative(g(x), x), (f(x) + 2*g(x))/x)] sol1 = [Eq(f(x), -C1*x + C2*x**3), Eq(g(x), C1*x + C2*x**3)] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) # Type 6 eqs2 = [Eq(Derivative(f(x), x), (2*f(x) + g(x) + 1)/x), Eq(Derivative(g(x), x), (x + f(x) + 2*g(x))/x)] sol2 = [Eq(f(x), C2*x**3 - x*(C1 + Rational(1, 4)) + x*log(x)*Rational(-1, 2) + Rational(-2, 3)), Eq(g(x), C2*x**3 + x*log(x)/2 + x*(C1 + Rational(-1, 4)) + Rational(1, 3))] assert dsolve(eqs2) == sol2 assert checksysodesol(eqs2, sol2) == (True, [0, 0]) def test_higher_order_to_first_order(): f, g = symbols('f g', cls=Function) x = symbols('x') eqs1 = [Eq(Derivative(f(x), (x, 2)), 2*f(x) + g(x)), Eq(Derivative(g(x), (x, 2)), -f(x))] sol1 = [Eq(f(x), -C2*x*exp(-x) + C3*x*exp(x) - (C1 - C2)*exp(-x) + (C3 + C4)*exp(x)), Eq(g(x), C2*x*exp(-x) - C3*x*exp(x) + (C1 + C2)*exp(-x) + (C3 - C4)*exp(x))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) eqs2 = [Eq(f(x).diff(x, 2), 0), Eq(g(x).diff(x, 2), f(x))] sol2 = [Eq(f(x), C1 + C2*x), Eq(g(x), C1*x**2/2 + C2*x**3/6 + C3 + C4*x)] assert dsolve(eqs2) == sol2 assert checksysodesol(eqs2, sol2) == (True, [0, 0]) eqs3 = [Eq(Derivative(f(x), (x, 2)), 2*f(x)), Eq(Derivative(g(x), (x, 2)), -f(x) + 2*g(x))] sol3 = [Eq(f(x), 4*C1*exp(-sqrt(2)*x) + 4*C2*exp(sqrt(2)*x)), Eq(g(x), sqrt(2)*C1*x*exp(-sqrt(2)*x) - sqrt(2)*C2*x*exp(sqrt(2)*x) + (C1 + sqrt(2)*C4)*exp(-sqrt(2)*x) + (C2 - sqrt(2)*C3)*exp(sqrt(2)*x))] assert dsolve(eqs3) == sol3 assert checksysodesol(eqs3, sol3) == (True, [0, 0]) eqs4 = [Eq(Derivative(f(x), (x, 2)), 2*f(x) + g(x)), Eq(Derivative(g(x), (x, 2)), 2*g(x))] sol4 = [Eq(f(x), C1*x*exp(sqrt(2)*x)/4 + C3*x*exp(-sqrt(2)*x)/4 + (C2/4 + sqrt(2)*C3/8)*exp(-sqrt(2)*x) - exp(sqrt(2)*x)*(sqrt(2)*C1/8 + C4*Rational(-1, 4))), Eq(g(x), sqrt(2)*C1*exp(sqrt(2)*x)/2 + sqrt(2)*C3*exp(-sqrt(2)*x)*Rational(-1, 2))] assert dsolve(eqs4) == sol4 assert checksysodesol(eqs4, sol4) == (True, [0, 0]) eqs5 = [Eq(f(x).diff(x, 2), f(x)), Eq(g(x).diff(x, 2), f(x))] sol5 = [Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), -C1*exp(-x) + C2*exp(x) + C3 + C4*x)] assert dsolve(eqs5) == sol5 assert checksysodesol(eqs5, sol5) == (True, [0, 0]) eqs6 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x)), Eq(Derivative(g(x), (x, 2)), -f(x) - g(x))] sol6 = [Eq(f(x), C1 + C2*x**2/2 + C2 + C4*x**3/6 + x*(C3 + C4)), Eq(g(x), -C1 + C2*x**2*Rational(-1, 2) - C3*x + C4*x**3*Rational(-1, 6))] assert dsolve(eqs6) == sol6 assert checksysodesol(eqs6, sol6) == (True, [0, 0]) eqs7 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x) + 1), Eq(Derivative(g(x), (x, 2)), f(x) + g(x) + 1)] sol7 = [Eq(f(x), -C1 - C2*x + sqrt(2)*C3*exp(sqrt(2)*x)/2 + sqrt(2)*C4*exp(-sqrt(2)*x)*Rational(-1, 2) + Rational(-1, 2)), Eq(g(x), C1 + C2*x + sqrt(2)*C3*exp(sqrt(2)*x)/2 + sqrt(2)*C4*exp(-sqrt(2)*x)*Rational(-1, 2) + Rational(-1, 2))] assert dsolve(eqs7) == sol7 assert checksysodesol(eqs7, sol7) == (True, [0, 0]) eqs8 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x) + 1), Eq(Derivative(g(x), (x, 2)), -f(x) - g(x) + 1)] sol8 = [Eq(f(x), C1 + C2 + C4*x**3/6 + x**4/12 + x**2*(C2/2 + Rational(1, 2)) + x*(C3 + C4)), Eq(g(x), -C1 - C3*x + C4*x**3*Rational(-1, 6) + x**4*Rational(-1, 12) - x**2*(C2/2 + Rational(-1, 2)))] assert dsolve(eqs8) == sol8 assert checksysodesol(eqs8, sol8) == (True, [0, 0]) x, y = symbols('x, y', cls=Function) t, l = symbols('t, l') eqs10 = [Eq(Derivative(x(t), (t, 2)), 5*x(t) + 43*y(t)), Eq(Derivative(y(t), (t, 2)), x(t) + 9*y(t))] sol10 = [Eq(x(t), C1*(61 - 9*sqrt(47))*sqrt(sqrt(47) + 7)*exp(-t*sqrt(sqrt(47) + 7))/2 + C2*sqrt(7 - sqrt(47))*(61 + 9*sqrt(47))*exp(-t*sqrt(7 - sqrt(47)))/2 + C3*(61 - 9*sqrt(47))*sqrt(sqrt(47) + 7)*exp(t*sqrt(sqrt(47) + 7))*Rational(-1, 2) + C4*sqrt(7 - sqrt(47))*(61 + 9*sqrt(47))*exp(t*sqrt(7 - sqrt(47)))*Rational(-1, 2)), Eq(y(t), C1*(7 - sqrt(47))*sqrt(sqrt(47) + 7)*exp(-t*sqrt(sqrt(47) + 7))*Rational(-1, 2) + C2*sqrt(7 - sqrt(47))*(sqrt(47) + 7)*exp(-t*sqrt(7 - sqrt(47)))*Rational(-1, 2) + C3*(7 - sqrt(47))*sqrt(sqrt(47) + 7)*exp(t*sqrt(sqrt(47) + 7))/2 + C4*sqrt(7 - sqrt(47))*(sqrt(47) + 7)*exp(t*sqrt(7 - sqrt(47)))/2)] assert dsolve(eqs10) == sol10 assert checksysodesol(eqs10, sol10) == (True, [0, 0]) eqs11 = [Eq(7*x(t) + Derivative(x(t), (t, 2)) - 9*Derivative(y(t), t), 0), Eq(7*y(t) + 9*Derivative(x(t), t) + Derivative(y(t), (t, 2)), 0)] sol11 = [Eq(y(t), C1*(9 - sqrt(109))*sin(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)/14 + C2*(9 - sqrt(109))*cos(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C3*(9 + sqrt(109))*sin(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)/14 + C4*(9 + sqrt(109))*cos(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)*Rational(-1, 14)), Eq(x(t), C1*(9 - sqrt(109))*cos(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C2*(9 - sqrt(109))*sin(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C3*(9 + sqrt(109))*cos(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)/14 + C4*(9 + sqrt(109))*sin(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)/14)] assert dsolve(eqs11) == sol11 assert checksysodesol(eqs11, sol11) == (True, [0, 0]) # Euler Systems # Note: To add examples of euler systems solver with non-homogeneous term. eqs13 = [Eq(Derivative(f(t), (t, 2)), Derivative(f(t), t)/t + f(t)/t**2 + g(t)/t**2), Eq(Derivative(g(t), (t, 2)), g(t)/t**2)] sol13 = [Eq(f(t), C1*(sqrt(5) + 3)*Rational(-1, 2)*t**(Rational(1, 2) + sqrt(5)*Rational(-1, 2)) + C2*t**(Rational(1, 2) + sqrt(5)/2)*(3 - sqrt(5))*Rational(-1, 2) - C3*t**(1 - sqrt(2))*(1 + sqrt(2)) - C4*t**(1 + sqrt(2))*(1 - sqrt(2))), Eq(g(t), C1*(1 + sqrt(5))*Rational(-1, 2)*t**(Rational(1, 2) + sqrt(5)*Rational(-1, 2)) + C2*t**(Rational(1, 2) + sqrt(5)/2)*(1 - sqrt(5))*Rational(-1, 2))] assert dsolve(eqs13) == sol13 assert checksysodesol(eqs13, sol13) == (True, [0, 0]) # Solving systems using dsolve separately eqs14 = [Eq(Derivative(f(t), (t, 2)), t*f(t)), Eq(Derivative(g(t), (t, 2)), t*g(t))] sol14 = [Eq(f(t), C1*airyai(t) + C2*airybi(t)), Eq(g(t), C3*airyai(t) + C4*airybi(t))] assert dsolve(eqs14) == sol14 assert checksysodesol(eqs14, sol14) == (True, [0, 0]) eqs15 = [Eq(Derivative(x(t), (t, 2)), t*(4*Derivative(x(t), t) + 8*Derivative(y(t), t))), Eq(Derivative(y(t), (t, 2)), t*(12*Derivative(x(t), t) - 6*Derivative(y(t), t)))] sol15 = [Eq(x(t), C1 - erf(sqrt(6)*t)*(sqrt(6)*sqrt(pi)*C2/33 + sqrt(6)*sqrt(pi)*C3*Rational(-1, 44)) + erfi(sqrt(5)*t)*(sqrt(5)*sqrt(pi)*C2*Rational(2, 55) + sqrt(5)*sqrt(pi)*C3*Rational(4, 55))), Eq(y(t), C4 + erf(sqrt(6)*t)*(sqrt(6)*sqrt(pi)*C2*Rational(2, 33) + sqrt(6)*sqrt(pi)*C3*Rational(-1, 22)) + erfi(sqrt(5)*t)*(sqrt(5)*sqrt(pi)*C2*Rational(3, 110) + sqrt(5)*sqrt(pi)*C3*Rational(3, 55)))] assert dsolve(eqs15) == sol15 assert checksysodesol(eqs15, sol15) == (True, [0, 0]) @slow def test_higher_order_to_first_order_9(): f, g = symbols('f g', cls=Function) x = symbols('x') eqs9 = [f(x) + g(x) - 2*exp(I*x) + 2*Derivative(f(x), x) + Derivative(f(x), (x, 2)), f(x) + g(x) - 2*exp(I*x) + 2*Derivative(g(x), x) + Derivative(g(x), (x, 2))] sol9 = [Eq(f(x), -C1 + C4*exp(-2*x)/2 - (C2/2 - C3/2)*exp(-x)*cos(x) + (C2/2 + C3/2)*exp(-x)*sin(x) + 2*((1 - 2*I)*exp(I*x)*sin(x)**2/5) + 2*((1 - 2*I)*exp(I*x)*cos(x)**2/5)), Eq(g(x), C1 - C4*exp(-2*x)/2 - (C2/2 - C3/2)*exp(-x)*cos(x) + (C2/2 + C3/2)*exp(-x)*sin(x) + 2*((1 - 2*I)*exp(I*x)*sin(x)**2/5) + 2*((1 - 2*I)*exp(I*x)*cos(x)**2/5))] assert dsolve(eqs9) == sol9 assert checksysodesol(eqs9, sol9) == (True, [0, 0]) def test_higher_order_to_first_order_12(): f, g = symbols('f g', cls=Function) x = symbols('x') x, y = symbols('x, y', cls=Function) t, l = symbols('t, l') eqs12 = [Eq(4*x(t) + Derivative(x(t), (t, 2)) + 8*Derivative(y(t), t), 0), Eq(4*y(t) - 8*Derivative(x(t), t) + Derivative(y(t), (t, 2)), 0)] sol12 = [Eq(y(t), C1*(2 - sqrt(5))*sin(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C2*(2 - sqrt(5))*cos(2*t*sqrt(4*sqrt(5) + 9))/2 + C3*(2 + sqrt(5))*sin(2*t*sqrt(9 - 4*sqrt(5)))*Rational(-1, 2) + C4*(2 + sqrt(5))*cos(2*t*sqrt(9 - 4*sqrt(5)))/2), Eq(x(t), C1*(2 - sqrt(5))*cos(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C2*(2 - sqrt(5))*sin(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C3*(2 + sqrt(5))*cos(2*t*sqrt(9 - 4*sqrt(5)))/2 + C4*(2 + sqrt(5))*sin(2*t*sqrt(9 - 4*sqrt(5)))/2)] assert dsolve(eqs12) == sol12 assert checksysodesol(eqs12, sol12) == (True, [0, 0]) def test_second_order_to_first_order_2(): f, g = symbols("f g", cls=Function) x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m") eqs2 = [Eq(f(x).diff(x, 2), 2*(x*g(x).diff(x) - g(x))), Eq(g(x).diff(x, 2),-2*(x*f(x).diff(x) - f(x)))] sol2 = [Eq(f(x), C1*x + x*Integral(C2*exp(-x_)*exp(I*exp(2*x_))/2 + C2*exp(-x_)*exp(-I*exp(2*x_))/2 - I*C3*exp(-x_)*exp(I*exp(2*x_))/2 + I*C3*exp(-x_)*exp(-I*exp(2*x_))/2, (x_, log(x)))), Eq(g(x), C4*x + x*Integral(I*C2*exp(-x_)*exp(I*exp(2*x_))/2 - I*C2*exp(-x_)*exp(-I*exp(2*x_))/2 + C3*exp(-x_)*exp(I*exp(2*x_))/2 + C3*exp(-x_)*exp(-I*exp(2*x_))/2, (x_, log(x))))] # XXX: dsolve hangs for this in integration assert dsolve_system(eqs2, simplify=False, doit=False) == [sol2] assert checksysodesol(eqs2, sol2) == (True, [0, 0]) eqs3 = (Eq(diff(f(t),t,t), 9*t*diff(g(t),t)-9*g(t)), Eq(diff(g(t),t,t),7*t*diff(f(t),t)-7*f(t))) sol3 = [Eq(f(t), C1*t + t*Integral(C2*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/2 + C2*exp(-t_)* exp(-3*sqrt(7)*exp(2*t_)/2)/2 + 3*sqrt(7)*C3*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/14 - 3*sqrt(7)*C3*exp(-t_)*exp(-3*sqrt(7)*exp(2*t_)/2)/14, (t_, log(t)))), Eq(g(t), C4*t + t*Integral(sqrt(7)*C2*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/6 - sqrt(7)*C2*exp(-t_)* exp(-3*sqrt(7)*exp(2*t_)/2)/6 + C3*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/2 + C3*exp(-t_)*exp(-3*sqrt(7)* exp(2*t_)/2)/2, (t_, log(t))))] # XXX: dsolve hangs for this in integration assert dsolve_system(eqs3, simplify=False, doit=False) == [sol3] assert checksysodesol(eqs3, sol3) == (True, [0, 0]) # Regression Test case for sympy#19238 # https://github.com/sympy/sympy/issues/19238 # Note: When the doit method is removed, these particular types of systems # can be divided first so that we have lesser number of big matrices. eqs5 = [Eq(Derivative(g(t), (t, 2)), a*m), Eq(Derivative(f(t), (t, 2)), 0)] sol5 = [Eq(g(t), C1 + C2*t + a*m*t**2/2), Eq(f(t), C3 + C4*t)] assert dsolve(eqs5) == sol5 assert checksysodesol(eqs5, sol5) == (True, [0, 0]) # Type 2 eqs6 = [Eq(Derivative(f(t), (t, 2)), f(t)/t**4), Eq(Derivative(g(t), (t, 2)), d*g(t)/t**4)] sol6 = [Eq(f(t), C1*sqrt(t**2)*exp(-1/t) - C2*sqrt(t**2)*exp(1/t)), Eq(g(t), C3*sqrt(t**2)*exp(-sqrt(d)/t)*d**Rational(-1, 2) - C4*sqrt(t**2)*exp(sqrt(d)/t)*d**Rational(-1, 2))] assert dsolve(eqs6) == sol6 assert checksysodesol(eqs6, sol6) == (True, [0, 0]) @slow def test_second_order_to_first_order_slow1(): f, g = symbols("f g", cls=Function) x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m") # Type 1 eqs1 = [Eq(f(x).diff(x, 2), 2/x *(x*g(x).diff(x) - g(x))), Eq(g(x).diff(x, 2),-2/x *(x*f(x).diff(x) - f(x)))] sol1 = [Eq(f(x), C1*x + 2*C2*x*Ci(2*x) - C2*sin(2*x) - 2*C3*x*Si(2*x) - C3*cos(2*x)), Eq(g(x), -2*C2*x*Si(2*x) - C2*cos(2*x) - 2*C3*x*Ci(2*x) + C3*sin(2*x) + C4*x)] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) def test_second_order_to_first_order_slow4(): f, g = symbols("f g", cls=Function) x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m") eqs4 = [Eq(Derivative(f(t), (t, 2)), t*sin(t)*Derivative(g(t), t) - g(t)*sin(t)), Eq(Derivative(g(t), (t, 2)), t*sin(t)*Derivative(f(t), t) - f(t)*sin(t))] sol4 = [Eq(f(t), C1*t + t*Integral(C2*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*exp(-sin(exp(t_)))/2 + C2*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2 - C3*exp(-t_)*exp(exp(t_)*cos(exp(t_)))* exp(-sin(exp(t_)))/2 + C3*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2, (t_, log(t)))), Eq(g(t), C4*t + t*Integral(-C2*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*exp(-sin(exp(t_)))/2 + C2*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2 + C3*exp(-t_)*exp(exp(t_)*cos(exp(t_)))* exp(-sin(exp(t_)))/2 + C3*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2, (t_, log(t))))] # XXX: dsolve hangs for this in integration assert dsolve_system(eqs4, simplify=False, doit=False) == [sol4] assert checksysodesol(eqs4, sol4) == (True, [0, 0]) def test_component_division(): f, g, h, k = symbols('f g h k', cls=Function) x = symbols("x") funcs = [f(x), g(x), h(x), k(x)] eqs1 = [Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), f(x)), Eq(Derivative(h(x), x), h(x)), Eq(Derivative(k(x), x), h(x)**4 + k(x))] sol1 = [Eq(f(x), 2*C1*exp(2*x)), Eq(g(x), C1*exp(2*x) + C2), Eq(h(x), C3*exp(x)), Eq(k(x), C3**4*exp(4*x)/3 + C4*exp(x))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0, 0, 0]) components1 = {((Eq(Derivative(f(x), x), 2*f(x)),), (Eq(Derivative(g(x), x), f(x)),)), ((Eq(Derivative(h(x), x), h(x)),), (Eq(Derivative(k(x), x), h(x)**4 + k(x)),))} eqsdict1 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {h(x)}}, {f(x): Eq(Derivative(f(x), x), 2*f(x)), g(x): Eq(Derivative(g(x), x), f(x)), h(x): Eq(Derivative(h(x), x), h(x)), k(x): Eq(Derivative(k(x), x), h(x)**4 + k(x))}) graph1 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), h(x))}] assert {tuple(tuple(scc) for scc in wcc) for wcc in _component_division(eqs1, funcs, x)} == components1 assert _eqs2dict(eqs1, funcs) == eqsdict1 assert [set(element) for element in _dict2graph(eqsdict1[0])] == graph1 eqs2 = [Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), f(x)), Eq(Derivative(h(x), x), h(x)), Eq(Derivative(k(x), x), f(x)**4 + k(x))] sol2 = [Eq(f(x), C1*exp(2*x)), Eq(g(x), C1*exp(2*x)/2 + C2), Eq(h(x), C3*exp(x)), Eq(k(x), C1**4*exp(8*x)/7 + C4*exp(x))] assert dsolve(eqs2) == sol2 assert checksysodesol(eqs2, sol2) == (True, [0, 0, 0, 0]) components2 = {frozenset([(Eq(Derivative(f(x), x), 2*f(x)),), (Eq(Derivative(g(x), x), f(x)),), (Eq(Derivative(k(x), x), f(x)**4 + k(x)),)]), frozenset([(Eq(Derivative(h(x), x), h(x)),)])} eqsdict2 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, {f(x): Eq(Derivative(f(x), x), 2*f(x)), g(x): Eq(Derivative(g(x), x), f(x)), h(x): Eq(Derivative(h(x), x), h(x)), k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) graph2 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), f(x))}] assert {frozenset(tuple(scc) for scc in wcc) for wcc in _component_division(eqs2, funcs, x)} == components2 assert _eqs2dict(eqs2, funcs) == eqsdict2 assert [set(element) for element in _dict2graph(eqsdict2[0])] == graph2 eqs3 = [Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), x + f(x)), Eq(Derivative(h(x), x), h(x)), Eq(Derivative(k(x), x), f(x)**4 + k(x))] sol3 = [Eq(f(x), C1*exp(2*x)), Eq(g(x), C1*exp(2*x)/2 + C2 + x**2/2), Eq(h(x), C3*exp(x)), Eq(k(x), C1**4*exp(8*x)/7 + C4*exp(x))] assert dsolve(eqs3) == sol3 assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0, 0]) components3 = {frozenset([(Eq(Derivative(f(x), x), 2*f(x)),), (Eq(Derivative(g(x), x), x + f(x)),), (Eq(Derivative(k(x), x), f(x)**4 + k(x)),)]), frozenset([(Eq(Derivative(h(x), x), h(x)),),])} eqsdict3 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, {f(x): Eq(Derivative(f(x), x), 2*f(x)), g(x): Eq(Derivative(g(x), x), x + f(x)), h(x): Eq(Derivative(h(x), x), h(x)), k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) graph3 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), f(x))}] assert {frozenset(tuple(scc) for scc in wcc) for wcc in _component_division(eqs3, funcs, x)} == components3 assert _eqs2dict(eqs3, funcs) == eqsdict3 assert [set(l) for l in _dict2graph(eqsdict3[0])] == graph3 # Note: To be uncommented when the default option to call dsolve first for # single ODE system can be rearranged. This can be done after the doit # option in dsolve is made False by default. eqs4 = [Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), Eq(Derivative(g(x), x), f(x) + x*g(x) + x), Eq(Derivative(h(x), x), h(x)), Eq(Derivative(k(x), x), f(x)**4 + k(x))] sol4 = [Eq(f(x), (C1/2 - sqrt(2)*C2/2 - sqrt(2)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 +\ sqrt(2)*x)/2, x)/2 + Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 +\ sqrt(2)*x)/2, x)/2)*exp(x**2/2 - sqrt(2)*x) + (C1/2 + sqrt(2)*C2/2 + sqrt(2)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2)*exp(x**2/2 + sqrt(2)*x)), Eq(g(x), (-sqrt(2)*C1/4 + C2/2 + Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 -\ sqrt(2)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/4)*exp(x**2/2 - sqrt(2)*x) + (sqrt(2)*C1/4 + C2/2 + Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + sqrt(2)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/4)*exp(x**2/2 + sqrt(2)*x)), Eq(h(x), C3*exp(x)), Eq(k(x), C4*exp(x) + exp(x)*Integral((C1*exp(x**2/2 - sqrt(2)*x)/2 + C1*exp(x**2/2 + sqrt(2)*x)/2 - sqrt(2)*C2*exp(x**2/2 - sqrt(2)*x)/2 + sqrt(2)*C2*exp(x**2/2 + sqrt(2)*x)/2 - sqrt(2)*exp(x**2/2 - sqrt(2)*x)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + exp(x**2/2 - sqrt(2)*x)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + sqrt(2)*exp(x**2/2 + sqrt(2)*x)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + exp(x**2/2 + sqrt(2)*x)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2)**4*exp(-x), x))] components4 = {(frozenset([Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), Eq(Derivative(g(x), x), x*g(x) + x + f(x))]), frozenset([Eq(Derivative(k(x), x), f(x)**4 + k(x)),])), (frozenset([Eq(Derivative(h(x), x), h(x)),]),)} eqsdict4 = ({f(x): {g(x)}, g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, {f(x): Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), g(x): Eq(Derivative(g(x), x), x*g(x) + x + f(x)), h(x): Eq(Derivative(h(x), x), h(x)), k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) graph4 = [{f(x), g(x), h(x), k(x)}, {(f(x), g(x)), (g(x), f(x)), (k(x), f(x))}] assert {tuple(frozenset(scc) for scc in wcc) for wcc in _component_division(eqs4, funcs, x)} == components4 assert _eqs2dict(eqs4, funcs) == eqsdict4 assert [set(element) for element in _dict2graph(eqsdict4[0])] == graph4 # XXX: dsolve hangs in integration here: assert dsolve_system(eqs4, simplify=False, doit=False) == [sol4] assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0, 0]) eqs5 = [Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), Eq(Derivative(g(x), x), x*g(x) + f(x)), Eq(Derivative(h(x), x), h(x)), Eq(Derivative(k(x), x), f(x)**4 + k(x))] sol5 = [Eq(f(x), (C1/2 - sqrt(2)*C2/2)*exp(x**2/2 - sqrt(2)*x) + (C1/2 + sqrt(2)*C2/2)*exp(x**2/2 + sqrt(2)*x)), Eq(g(x), (-sqrt(2)*C1/4 + C2/2)*exp(x**2/2 - sqrt(2)*x) + (sqrt(2)*C1/4 + C2/2)*exp(x**2/2 + sqrt(2)*x)), Eq(h(x), C3*exp(x)), Eq(k(x), C4*exp(x) + exp(x)*Integral((C1*exp(x**2/2 - sqrt(2)*x)/2 + C1*exp(x**2/2 + sqrt(2)*x)/2 - sqrt(2)*C2*exp(x**2/2 - sqrt(2)*x)/2 + sqrt(2)*C2*exp(x**2/2 + sqrt(2)*x)/2)**4*exp(-x), x))] components5 = {(frozenset([Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), Eq(Derivative(g(x), x), x*g(x) + f(x))]), frozenset([Eq(Derivative(k(x), x), f(x)**4 + k(x)),])), (frozenset([Eq(Derivative(h(x), x), h(x)),]),)} eqsdict5 = ({f(x): {g(x)}, g(x): {f(x)}, h(x): set(), k(x): {f(x)}}, {f(x): Eq(Derivative(f(x), x), x*f(x) + 2*g(x)), g(x): Eq(Derivative(g(x), x), x*g(x) + f(x)), h(x): Eq(Derivative(h(x), x), h(x)), k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))}) graph5 = [{f(x), g(x), h(x), k(x)}, {(f(x), g(x)), (g(x), f(x)), (k(x), f(x))}] assert {tuple(frozenset(scc) for scc in wcc) for wcc in _component_division(eqs5, funcs, x)} == components5 assert _eqs2dict(eqs5, funcs) == eqsdict5 assert [set(element) for element in _dict2graph(eqsdict5[0])] == graph5 # XXX: dsolve hangs in integration here: assert dsolve_system(eqs5, simplify=False, doit=False) == [sol5] assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0, 0]) def test_linodesolve(): t, x, a = symbols("t x a") f, g, h = symbols("f g h", cls=Function) # Testing the Errors raises(ValueError, lambda: linodesolve(1, t)) raises(ValueError, lambda: linodesolve(a, t)) A1 = Matrix([[1, 2], [2, 4], [4, 6]]) raises(NonSquareMatrixError, lambda: linodesolve(A1, t)) A2 = Matrix([[1, 2, 1], [3, 1, 2]]) raises(NonSquareMatrixError, lambda: linodesolve(A2, t)) # Testing auto functionality func = [f(t), g(t)] eq = [Eq(f(t).diff(t) + g(t).diff(t), g(t)), Eq(g(t).diff(t), f(t))] ceq = canonical_odes(eq, func, t) (A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1) A = A0 sol = [C1*(-Rational(1, 2) + sqrt(5)/2)*exp(t*(-Rational(1, 2) + sqrt(5)/2)) + C2*(-sqrt(5)/2 - Rational(1, 2))* exp(t*(-sqrt(5)/2 - Rational(1, 2))), C1*exp(t*(-Rational(1, 2) + sqrt(5)/2)) + C2*exp(t*(-sqrt(5)/2 - Rational(1, 2)))] assert constant_renumber(linodesolve(A, t), variables=Tuple(*eq).free_symbols) == sol # Testing the Errors raises(ValueError, lambda: linodesolve(1, t, b=Matrix([t+1]))) raises(ValueError, lambda: linodesolve(a, t, b=Matrix([log(t) + sin(t)]))) raises(ValueError, lambda: linodesolve(Matrix([7]), t, b=t**2)) raises(ValueError, lambda: linodesolve(Matrix([a+10]), t, b=log(t)*cos(t))) raises(ValueError, lambda: linodesolve(7, t, b=t**2)) raises(ValueError, lambda: linodesolve(a, t, b=log(t) + sin(t))) A1 = Matrix([[1, 2], [2, 4], [4, 6]]) b1 = Matrix([t, 1, t**2]) raises(NonSquareMatrixError, lambda: linodesolve(A1, t, b=b1)) A2 = Matrix([[1, 2, 1], [3, 1, 2]]) b2 = Matrix([t, t**2]) raises(NonSquareMatrixError, lambda: linodesolve(A2, t, b=b2)) raises(ValueError, lambda: linodesolve(A1[:2, :], t, b=b1)) raises(ValueError, lambda: linodesolve(A1[:2, :], t, b=b1[:1])) # DOIT check A1 = Matrix([[1, -1], [1, -1]]) b1 = Matrix([15*t - 10, -15*t - 5]) sol1 = [C1 + C2*t + C2 - 10*t**3 + 10*t**2 + t*(15*t**2 - 5*t) - 10*t, C1 + C2*t - 10*t**3 - 5*t**2 + t*(15*t**2 - 5*t) - 5*t] assert constant_renumber(linodesolve(A1, t, b=b1, type="type2", doit=True), variables=[t]) == sol1 # Testing auto functionality func = [f(t), g(t)] eq = [Eq(f(t).diff(t) + g(t).diff(t), g(t) + t), Eq(g(t).diff(t), f(t))] ceq = canonical_odes(eq, func, t) (A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1) A = A0 sol = [-C1*exp(-t/2 + sqrt(5)*t/2)/2 + sqrt(5)*C1*exp(-t/2 + sqrt(5)*t/2)/2 - sqrt(5)*C2*exp(-sqrt(5)*t/2 - t/2)/2 - C2*exp(-sqrt(5)*t/2 - t/2)/2 - exp(-t/2 + sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)), t)/2 + sqrt(5)*exp(-t/2 + sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)), t)/2 - sqrt(5)*exp(-sqrt(5)*t/2 - t/2)*Integral(-sqrt(5)*t*exp(t/2 + sqrt(5)*t/2)/5, t)/2 - exp(-sqrt(5)*t/2 - t/2)*Integral(-sqrt(5)*t*exp(t/2 + sqrt(5)*t/2)/5, t)/2, C1*exp(-t/2 + sqrt(5)*t/2) + C2*exp(-sqrt(5)*t/2 - t/2) + exp(-t/2 + sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)), t) + exp(-sqrt(5)*t/2 - t/2)*Integral(-sqrt(5)*t*exp(t/2 + sqrt(5)*t/2)/5, t)] assert constant_renumber(linodesolve(A, t, b=b), variables=[t]) == sol # non-homogeneous term assumed to be 0 sol1 = [-C1*exp(-t/2 + sqrt(5)*t/2)/2 + sqrt(5)*C1*exp(-t/2 + sqrt(5)*t/2)/2 - sqrt(5)*C2*exp(-sqrt(5)*t/2 - t/2)/2 - C2*exp(-sqrt(5)*t/2 - t/2)/2, C1*exp(-t/2 + sqrt(5)*t/2) + C2*exp(-sqrt(5)*t/2 - t/2)] assert constant_renumber(linodesolve(A, t, type="type2"), variables=[t]) == sol1 # Testing the Errors raises(ValueError, lambda: linodesolve(t+10, t)) raises(ValueError, lambda: linodesolve(a*t, t)) A1 = Matrix([[1, t], [-t, 1]]) B1, _ = _is_commutative_anti_derivative(A1, t) raises(NonSquareMatrixError, lambda: linodesolve(A1[:, :1], t, B=B1)) raises(ValueError, lambda: linodesolve(A1, t, B=1)) A2 = Matrix([[t, t, t], [t, t, t], [t, t, t]]) B2, _ = _is_commutative_anti_derivative(A2, t) raises(NonSquareMatrixError, lambda: linodesolve(A2, t, B=B2[:2, :])) raises(ValueError, lambda: linodesolve(A2, t, B=2)) raises(ValueError, lambda: linodesolve(A2, t, B=B2, type="type31")) raises(ValueError, lambda: linodesolve(A1, t, B=B2)) raises(ValueError, lambda: linodesolve(A2, t, B=B1)) # Testing auto functionality func = [f(t), g(t)] eq = [Eq(f(t).diff(t), f(t) + t*g(t)), Eq(g(t).diff(t), -t*f(t) + g(t))] ceq = canonical_odes(eq, func, t) (A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1) A = A0 sol = [(C1/2 - I*C2/2)*exp(I*t**2/2 + t) + (C1/2 + I*C2/2)*exp(-I*t**2/2 + t), (-I*C1/2 + C2/2)*exp(-I*t**2/2 + t) + (I*C1/2 + C2/2)*exp(I*t**2/2 + t)] assert constant_renumber(linodesolve(A, t), variables=Tuple(*eq).free_symbols) == sol assert constant_renumber(linodesolve(A, t, type="type3"), variables=Tuple(*eq).free_symbols) == sol A1 = Matrix([[t, 1], [t, -1]]) raises(NotImplementedError, lambda: linodesolve(A1, t)) # Testing the Errors raises(ValueError, lambda: linodesolve(t+10, t, b=Matrix([t+1]))) raises(ValueError, lambda: linodesolve(a*t, t, b=Matrix([log(t) + sin(t)]))) raises(ValueError, lambda: linodesolve(Matrix([7*t]), t, b=t**2)) raises(ValueError, lambda: linodesolve(Matrix([a + 10*log(t)]), t, b=log(t)*cos(t))) raises(ValueError, lambda: linodesolve(7*t, t, b=t**2)) raises(ValueError, lambda: linodesolve(a*t**2, t, b=log(t) + sin(t))) A1 = Matrix([[1, t], [-t, 1]]) b1 = Matrix([t, t ** 2]) B1, _ = _is_commutative_anti_derivative(A1, t) raises(NonSquareMatrixError, lambda: linodesolve(A1[:, :1], t, b=b1)) A2 = Matrix([[t, t, t], [t, t, t], [t, t, t]]) b2 = Matrix([t, 1, t**2]) B2, _ = _is_commutative_anti_derivative(A2, t) raises(NonSquareMatrixError, lambda: linodesolve(A2[:2, :], t, b=b2)) raises(ValueError, lambda: linodesolve(A1, t, b=b2)) raises(ValueError, lambda: linodesolve(A2, t, b=b1)) raises(ValueError, lambda: linodesolve(A1, t, b=b1, B=B2)) raises(ValueError, lambda: linodesolve(A2, t, b=b2, B=B1)) # Testing auto functionality func = [f(x), g(x), h(x)] eq = [Eq(f(x).diff(x), x*(f(x) + g(x) + h(x)) + x), Eq(g(x).diff(x), x*(f(x) + g(x) + h(x)) + x), Eq(h(x).diff(x), x*(f(x) + g(x) + h(x)) + 1)] ceq = canonical_odes(eq, func, x) (A1, A0), b = linear_ode_to_matrix(ceq[0], func, x, 1) A = A0 _x1 = exp(-3*x**2/2) _x2 = exp(3*x**2/2) _x3 = Integral(2*_x1*x/3 + _x1/3 + x/3 - Rational(1, 3), x) _x4 = 2*_x2*_x3/3 _x5 = Integral(2*_x1*x/3 + _x1/3 - 2*x/3 + Rational(2, 3), x) sol = [ C1*_x2/3 - C1/3 + C2*_x2/3 - C2/3 + C3*_x2/3 + 2*C3/3 + _x2*_x5/3 + _x3/3 + _x4 - _x5/3, C1*_x2/3 + 2*C1/3 + C2*_x2/3 - C2/3 + C3*_x2/3 - C3/3 + _x2*_x5/3 + _x3/3 + _x4 - _x5/3, C1*_x2/3 - C1/3 + C2*_x2/3 + 2*C2/3 + C3*_x2/3 - C3/3 + _x2*_x5/3 - 2*_x3/3 + _x4 + 2*_x5/3, ] assert constant_renumber(linodesolve(A, x, b=b), variables=Tuple(*eq).free_symbols) == sol assert constant_renumber(linodesolve(A, x, b=b, type="type4"), variables=Tuple(*eq).free_symbols) == sol A1 = Matrix([[t, 1], [t, -1]]) raises(NotImplementedError, lambda: linodesolve(A1, t, b=b1)) # non-homogeneous term not passed sol1 = [-C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)] assert constant_renumber(linodesolve(A, x, type="type4", doit=True), variables=Tuple(*eq).free_symbols) == sol1 @slow def test_linear_3eq_order1_type4_slow(): x, y, z = symbols('x, y, z', cls=Function) t = Symbol('t') f = t ** 3 + log(t) g = t ** 2 + sin(t) eq1 = (Eq(diff(x(t), t), (4 * f + g) * x(t) - f * y(t) - 2 * f * z(t)), Eq(diff(y(t), t), 2 * f * x(t) + (f + g) * y(t) - 2 * f * z(t)), Eq(diff(z(t), t), 5 * f * x(t) + f * y( t) + (-3 * f + g) * z(t))) with dotprodsimp(True): dsolve(eq1) @slow def test_linear_neq_order1_type2_slow1(): i, r1, c1, r2, c2, t = symbols('i, r1, c1, r2, c2, t') x1 = Function('x1') x2 = Function('x2') eq1 = r1*c1*Derivative(x1(t), t) + x1(t) - x2(t) - r1*i eq2 = r2*c1*Derivative(x1(t), t) + r2*c2*Derivative(x2(t), t) + x2(t) - r2*i eq = [eq1, eq2] # XXX: Solution is too complicated [sol] = dsolve_system(eq, simplify=False, doit=False) assert checksysodesol(eq, sol) == (True, [0, 0]) # Regression test case for issue #9204 # https://github.com/sympy/sympy/issues/9204 @tooslow def test_linear_new_order1_type2_de_lorentz_slow_check(): m = Symbol("m", real=True) q = Symbol("q", real=True) t = Symbol("t", real=True) e1, e2, e3 = symbols("e1:4", real=True) b1, b2, b3 = symbols("b1:4", real=True) v1, v2, v3 = symbols("v1:4", cls=Function, real=True) eqs = [ -e1*q + m*Derivative(v1(t), t) - q*(-b2*v3(t) + b3*v2(t)), -e2*q + m*Derivative(v2(t), t) - q*(b1*v3(t) - b3*v1(t)), -e3*q + m*Derivative(v3(t), t) - q*(-b1*v2(t) + b2*v1(t)) ] sol = dsolve(eqs) assert checksysodesol(eqs, sol) == (True, [0, 0, 0]) # Regression test case for issue #14001 # https://github.com/sympy/sympy/issues/14001 @slow def test_linear_neq_order1_type2_slow_check(): RC, t, C, Vs, L, R1, V0, I0 = symbols("RC t C Vs L R1 V0 I0") V = Function("V") I = Function("I") system = [Eq(V(t).diff(t), -1/RC*V(t) + I(t)/C), Eq(I(t).diff(t), -R1/L*I(t) - 1/L*V(t) + Vs/L)] [sol] = dsolve_system(system, simplify=False, doit=False) assert checksysodesol(system, sol) == (True, [0, 0]) def _linear_3eq_order1_type4_long(): x, y, z = symbols('x, y, z', cls=Function) t = Symbol('t') f = t ** 3 + log(t) g = t ** 2 + sin(t) eq1 = (Eq(diff(x(t), t), (4*f + g)*x(t) - f*y(t) - 2*f*z(t)), Eq(diff(y(t), t), 2*f*x(t) + (f + g)*y(t) - 2*f*z(t)), Eq(diff(z(t), t), 5*f*x(t) + f*y( t) + (-3*f + g)*z(t))) dsolve_sol = dsolve(eq1) dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol] x_1 = sqrt(-t**6 - 8*t**3*log(t) + 8*t**3 - 16*log(t)**2 + 32*log(t) - 16) x_2 = sqrt(3) x_3 = 8324372644*C1*x_1*x_2 + 4162186322*C2*x_1*x_2 - 8324372644*C3*x_1*x_2 x_4 = 1 / (1903457163*t**3 + 3825881643*x_1*x_2 + 7613828652*log(t) - 7613828652) x_5 = exp(t**3/3 + t*x_1*x_2/4 - cos(t)) x_6 = exp(t**3/3 - t*x_1*x_2/4 - cos(t)) x_7 = exp(t**4/2 + t**3/3 + 2*t*log(t) - 2*t - cos(t)) x_8 = 91238*C1*x_1*x_2 + 91238*C2*x_1*x_2 - 91238*C3*x_1*x_2 x_9 = 1 / (66049*t**3 - 50629*x_1*x_2 + 264196*log(t) - 264196) x_10 = 50629 * C1 / 25189 + 37909*C2/25189 - 50629*C3/25189 - x_3*x_4 x_11 = -50629*C1/25189 - 12720*C2/25189 + 50629*C3/25189 + x_3*x_4 sol = [Eq(x(t), x_10*x_5 + x_11*x_6 + x_7*(C1 - C2)), Eq(y(t), x_10*x_5 + x_11*x_6), Eq(z(t), x_5*( -424*C1/257 - 167*C2/257 + 424*C3/257 - x_8*x_9) + x_6*(167*C1/257 + 424*C2/257 - 167*C3/257 + x_8*x_9) + x_7*(C1 - C2))] assert dsolve_sol1 == sol assert checksysodesol(eq1, dsolve_sol1) == (True, [0, 0, 0]) @slow def test_neq_order1_type4_slow_check1(): f, g = symbols("f g", cls=Function) x = symbols("x") eqs = [Eq(diff(f(x), x), x*f(x) + x**2*g(x) + x), Eq(diff(g(x), x), 2*x**2*f(x) + (x + 3*x**2)*g(x) + 1)] sol = dsolve(eqs) assert checksysodesol(eqs, sol) == (True, [0, 0]) @slow def test_neq_order1_type4_slow_check2(): f, g, h = symbols("f, g, h", cls=Function) x = Symbol("x") eqs = [ Eq(Derivative(f(x), x), x*h(x) + f(x) + g(x) + 1), Eq(Derivative(g(x), x), x*g(x) + f(x) + h(x) + 10), Eq(Derivative(h(x), x), x*f(x) + x + g(x) + h(x)) ] with dotprodsimp(True): sol = dsolve(eqs) assert checksysodesol(eqs, sol) == (True, [0, 0, 0]) def _neq_order1_type4_slow3(): f, g = symbols("f g", cls=Function) x = symbols("x") eqs = [ Eq(Derivative(f(x), x), x*f(x) + g(x) + sin(x)), Eq(Derivative(g(x), x), x**2 + x*g(x) - f(x)) ] sol = [ Eq(f(x), (C1/2 - I*C2/2 - I*Integral(x**2*exp(-x**2/2 - I*x)/2 + x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 - I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 + Integral(-I*x**2*exp(-x**2/2 - I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 + I*x) + (C1/2 + I*C2/2 + I*Integral(x**2*exp(-x**2/2 - I*x)/2 + x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 - I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 + Integral(-I*x**2*exp(-x**2/2 - I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 - I*x)), Eq(g(x), (-I*C1/2 + C2/2 + Integral(x**2*exp(-x**2/2 - I*x)/2 + x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 - I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 - I*Integral(-I*x**2*exp(-x**2/2 - I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 - I*x) + (I*C1/2 + C2/2 + Integral(x**2*exp(-x**2/2 - I*x)/2 + x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 - I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 + I*Integral(-I*x**2*exp(-x**2/2 - I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 + I*x)) ] return eqs, sol def test_neq_order1_type4_slow3(): eqs, sol = _neq_order1_type4_slow3() assert dsolve_system(eqs, simplify=False, doit=False) == [sol] # XXX: dsolve gives an error in integration: # assert dsolve(eqs) == sol # https://github.com/sympy/sympy/issues/20155 @slow def test_neq_order1_type4_slow_check3(): eqs, sol = _neq_order1_type4_slow3() assert checksysodesol(eqs, sol) == (True, [0, 0]) @tooslow @XFAIL def test_linear_3eq_order1_type4_long_dsolve_slow_xfail(): eq, sol = _linear_3eq_order1_type4_long() dsolve_sol = dsolve(eq) dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol] assert dsolve_sol1 == sol @tooslow def test_linear_3eq_order1_type4_long_dsolve_dotprodsimp(): eq, sol = _linear_3eq_order1_type4_long() # XXX: Only works with dotprodsimp see # test_linear_3eq_order1_type4_long_dsolve_slow_xfail which is too slow with dotprodsimp(True): dsolve_sol = dsolve(eq) dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol] assert dsolve_sol1 == sol @tooslow def test_linear_3eq_order1_type4_long_check(): eq, sol = _linear_3eq_order1_type4_long() assert checksysodesol(eq, sol) == (True, [0, 0, 0]) def test_dsolve_system(): f, g = symbols("f g", cls=Function) x = symbols("x") eqs = [Eq(f(x).diff(x), f(x) + g(x)), Eq(g(x).diff(x), f(x) + g(x))] funcs = [f(x), g(x)] sol = [[Eq(f(x), -C1 + C2*exp(2*x)), Eq(g(x), C1 + C2*exp(2*x))]] assert dsolve_system(eqs, funcs=funcs, t=x, doit=True) == sol raises(ValueError, lambda: dsolve_system(1)) raises(ValueError, lambda: dsolve_system(eqs, 1)) raises(ValueError, lambda: dsolve_system(eqs, funcs, 1)) raises(ValueError, lambda: dsolve_system(eqs, funcs[:1], x)) eq = (Eq(f(x).diff(x), 12 * f(x) - 6 * g(x)), Eq(g(x).diff(x) ** 2, 11 * f(x) + 3 * g(x))) raises(NotImplementedError, lambda: dsolve_system(eq) == ([], [])) raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)]) == ([], [])) raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], t=x) == ([], [])) raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], t=x, ics={f(0): 1, g(0): 1}) == ([], [])) raises(NotImplementedError, lambda: dsolve_system(eq, t=x, ics={f(0): 1, g(0): 1}) == ([], [])) raises(NotImplementedError, lambda: dsolve_system(eq, ics={f(0): 1, g(0): 1}) == ([], [])) raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], ics={f(0): 1, g(0): 1}) == ([], [])) def test_dsolve(): f, g = symbols('f g', cls=Function) x, y = symbols('x y') eqs = [f(x).diff(x) - x, f(x).diff(x) + x] with raises(ValueError): dsolve(eqs) eqs = [f(x, y).diff(x)] with raises(ValueError): dsolve(eqs) eqs = [f(x, y).diff(x)+g(x).diff(x), g(x).diff(x)] with raises(ValueError): dsolve(eqs) @slow def test_higher_order1_slow1(): x, y = symbols("x y", cls=Function) t = symbols("t") eq = [ Eq(diff(x(t),t,t), (log(t)+t**2)*diff(x(t),t)+(log(t)+t**2)*3*diff(y(t),t)), Eq(diff(y(t),t,t), (log(t)+t**2)*2*diff(x(t),t)+(log(t)+t**2)*9*diff(y(t),t)) ] sol, = dsolve_system(eq, simplify=False, doit=False) # The solution is too long to write out explicitly and checkodesol is too # slow so we test for particular values of t: for e in eq: res = (e.lhs - e.rhs).subs({sol[0].lhs:sol[0].rhs, sol[1].lhs:sol[1].rhs}) res = res.subs({d: d.doit(deep=False) for d in res.atoms(Derivative)}) assert ratsimp(res.subs(t, 1)) == 0 def test_second_order_type2_slow1(): x, y, z = symbols('x, y, z', cls=Function) t, l = symbols('t, l') eqs1 = [Eq(Derivative(x(t), (t, 2)), t*(2*x(t) + y(t))), Eq(Derivative(y(t), (t, 2)), t*(-x(t) + 2*y(t)))] sol1 = [Eq(x(t), I*C1*airyai(t*(2 - I)**(S(1)/3)) + I*C2*airybi(t*(2 - I)**(S(1)/3)) - I*C3*airyai(t*(2 + I)**(S(1)/3)) - I*C4*airybi(t*(2 + I)**(S(1)/3))), Eq(y(t), C1*airyai(t*(2 - I)**(S(1)/3)) + C2*airybi(t*(2 - I)**(S(1)/3)) + C3*airyai(t*(2 + I)**(S(1)/3)) + C4*airybi(t*(2 + I)**(S(1)/3)))] assert dsolve(eqs1) == sol1 assert checksysodesol(eqs1, sol1) == (True, [0, 0]) @tooslow @XFAIL def test_nonlinear_3eq_order1_type1(): a, b, c = symbols('a b c') eqs = [ a * f(x).diff(x) - (b - c) * g(x) * h(x), b * g(x).diff(x) - (c - a) * h(x) * f(x), c * h(x).diff(x) - (a - b) * f(x) * g(x), ] assert dsolve(eqs) # NotImplementedError @XFAIL def test_nonlinear_3eq_order1_type4(): eqs = [ Eq(f(x).diff(x), (2*h(x)*g(x) - 3*g(x)*h(x))), Eq(g(x).diff(x), (4*f(x)*h(x) - 2*h(x)*f(x))), Eq(h(x).diff(x), (3*g(x)*f(x) - 4*f(x)*g(x))), ] dsolve(eqs) # KeyError when matching # sol = ? # assert dsolve_sol == sol # assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0]) @tooslow @XFAIL def test_nonlinear_3eq_order1_type3(): eqs = [ Eq(f(x).diff(x), (2*f(x)**2 - 3 )), Eq(g(x).diff(x), (4 - 2*h(x) )), Eq(h(x).diff(x), (3*h(x) - 4*f(x)**2)), ] dsolve(eqs) # Not sure if this finishes... # sol = ? # assert dsolve_sol == sol # assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0]) @XFAIL def test_nonlinear_3eq_order1_type5(): eqs = [ Eq(f(x).diff(x), f(x)*(2*f(x) - 3*g(x))), Eq(g(x).diff(x), g(x)*(4*g(x) - 2*h(x))), Eq(h(x).diff(x), h(x)*(3*h(x) - 4*f(x))), ] dsolve(eqs) # KeyError # sol = ? # assert dsolve_sol == sol # assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0]) def test_linear_2eq_order1(): x, y, z = symbols('x, y, z', cls=Function) k, l, m, n = symbols('k, l, m, n', Integer=True) t = Symbol('t') x0, y0 = symbols('x0, y0', cls=Function) eq1 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23)) sol1 = [Eq(x(t), C1*exp(t*(sqrt(6) + 3)) + C2*exp(t*(-sqrt(6) + 3)) - Rational(22, 3)), \ Eq(y(t), C1*(2 + sqrt(6))*exp(t*(sqrt(6) + 3)) + C2*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) - Rational(5, 3))] assert checksysodesol(eq1, sol1) == (True, [0, 0]) eq2 = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23)) sol2 = [Eq(x(t), (C1*cos(sqrt(2)*t) + C2*sin(sqrt(2)*t))*exp(t) - Rational(58, 3)), \ Eq(y(t), (-sqrt(2)*C1*sin(sqrt(2)*t) + sqrt(2)*C2*cos(sqrt(2)*t))*exp(t) - Rational(185, 3))] assert checksysodesol(eq2, sol2) == (True, [0, 0]) eq3 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t))) sol3 = [Eq(x(t), (C1*exp(2*t) + C2*exp(-2*t))*exp(Rational(5, 2)*t**2)), \ Eq(y(t), (C1*exp(2*t) - C2*exp(-2*t))*exp(Rational(5, 2)*t**2))] assert checksysodesol(eq3, sol3) == (True, [0, 0]) eq4 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t))) sol4 = [Eq(x(t), (C1*cos((t**3)/3) + C2*sin((t**3)/3))*exp(Rational(5, 2)*t**2)), \ Eq(y(t), (-C1*sin((t**3)/3) + C2*cos((t**3)/3))*exp(Rational(5, 2)*t**2))] assert checksysodesol(eq4, sol4) == (True, [0, 0]) eq5 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t))) sol5 = [Eq(x(t), (C1*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \ C2*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2)), \ Eq(y(t), (C1*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \ C2*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2))] assert checksysodesol(eq5, sol5) == (True, [0, 0]) eq6 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), (1-t**2)*x(t) + (5*t+9*t**2)*y(t))) sol6 = [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), \ Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + \ exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))] s = dsolve(eq6) assert s == sol6 # too complicated to test with subs and simplify # assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this one fails def test_nonlinear_2eq_order1(): x, y, z = symbols('x, y, z', cls=Function) t = Symbol('t') eq1 = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5)) sol1 = [ Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] assert dsolve(eq1) == sol1 assert checksysodesol(eq1, sol1) == (True, [0, 0]) eq2 = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5)) sol2 = [ Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] assert dsolve(eq2) == sol2 assert checksysodesol(eq2, sol2) == (True, [0, 0]) eq3 = (Eq(diff(x(t),t), y(t)*x(t)), Eq(diff(y(t),t), x(t)**3)) tt = Rational(2, 3) sol3 = [ Eq(x(t), 6**tt/(6*(-sinh(sqrt(C1)*(C2 + t)/2)/sqrt(C1))**tt)), Eq(y(t), sqrt(C1 + C1/sinh(sqrt(C1)*(C2 + t)/2)**2)/3)] assert dsolve(eq3) == sol3 # FIXME: assert checksysodesol(eq3, sol3) == (True, [0, 0]) eq4 = (Eq(diff(x(t),t),x(t)*y(t)*sin(t)**2), Eq(diff(y(t),t),y(t)**2*sin(t)**2)) sol4 = {Eq(x(t), -2*exp(C1)/(C2*exp(C1) + t - sin(2*t)/2)), Eq(y(t), -2/(C1 + t - sin(2*t)/2))} assert dsolve(eq4) == sol4 # FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0]) eq5 = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2)) sol5 = {Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)} assert dsolve(eq5) == sol5 assert checksysodesol(eq5, sol5) == (True, [0, 0]) eq6 = (Eq(diff(x(t),t),x(t)**2*y(t)**3), Eq(diff(y(t),t),y(t)**5)) sol6 = [ Eq(x(t), 1/(C1 - 1/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), 1/(C1 + (-1/(4*C2 + 4*t))**(Rational(-1, 4)))), Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), 1/(C1 + I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)), Eq(x(t), 1/(C1 - I/(-1/(4*C2 + 4*t))**Rational(1, 4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))] assert dsolve(eq6) == sol6 assert checksysodesol(eq6, sol6) == (True, [0, 0]) @slow def test_nonlinear_3eq_order1(): x, y, z = symbols('x, y, z', cls=Function) t, u = symbols('t u') eq1 = (4*diff(x(t),t) + 2*y(t)*z(t), 3*diff(y(t),t) - z(t)*x(t), 5*diff(z(t),t) - x(t)*y(t)) sol1 = [Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), (u, x(t))), C3 - sqrt(15)*t/15), Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), (u, y(t))), C3 + sqrt(5)*t/10), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)* sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*t/6)] assert [i.dummy_eq(j) for i, j in zip(dsolve(eq1), sol1)] # FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0, 0]) eq2 = (4*diff(x(t),t) + 2*y(t)*z(t)*sin(t), 3*diff(y(t),t) - z(t)*x(t)*sin(t), 5*diff(z(t),t) - x(t)*y(t)*sin(t)) sol2 = [Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), (u, x(t))), C3 + sqrt(5)*cos(t)/10), Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), (u, y(t))), C3 - sqrt(15)*cos(t)/15), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)* sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*cos(t)/6)] assert [i.dummy_eq(j) for i, j in zip(dsolve(eq2), sol2)] # FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0, 0]) def test_C1_function_9239(): t = Symbol('t') C1 = Function('C1') C2 = Function('C2') C3 = Symbol('C3') C4 = Symbol('C4') eq = (Eq(diff(C1(t), t), 9*C2(t)), Eq(diff(C2(t), t), 12*C1(t))) sol = [Eq(C1(t), 9*C3*exp(6*sqrt(3)*t) + 9*C4*exp(-6*sqrt(3)*t)), Eq(C2(t), 6*sqrt(3)*C3*exp(6*sqrt(3)*t) - 6*sqrt(3)*C4*exp(-6*sqrt(3)*t))] assert checksysodesol(eq, sol) == (True, [0, 0]) def test_dsolve_linsystem_symbol(): eps = Symbol('epsilon', positive=True) eq1 = (Eq(diff(f(x), x), -eps*g(x)), Eq(diff(g(x), x), eps*f(x))) sol1 = [Eq(f(x), -C1*eps*cos(eps*x) - C2*eps*sin(eps*x)), Eq(g(x), -C1*eps*sin(eps*x) + C2*eps*cos(eps*x))] assert checksysodesol(eq1, sol1) == (True, [0, 0])