def test_checkodesol():
# For the most part, checkodesol is well tested in the tests below.
# These tests only handle cases not checked below.
raises(ValueError, lambda: checkodesol(f(x, y).diff(x), Eq(f(x, y), x)))
raises(ValueError,
lambda: checkodesol(f(x).diff(x), Eq(f(x, y), x), f(x, y)))
assert checkodesol(f(x).diff(x), Eq(f(x, y), x)) == \
(False, -f(x).diff(x) + f(x, y).diff(x) - 1)
assert checkodesol(f(x).diff(x), Eq(f(x), x)) is not True
assert checkodesol(f(x).diff(x), Eq(f(x), x)) == (False, 1)
sol1 = Eq(f(x)**5 + 11 * f(x) - 2 * f(x) + x, 0)
assert checkodesol(diff(sol1.lhs, x), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x) * exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 2), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 2) * exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3) * exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3), Eq(f(x), x*log(x))) == \
(False, 60*x**4*((log(x) + 1)**2 + log(x))*(
log(x) + 1)*log(x)**2 - 5*x**4*log(x)**4 - 9)
assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x, 0)) == \
(True, 0)
assert checkodesol(diff(exp(f(x)) + x, x) * x,
Eq(exp(f(x)) + x, 0),
solve_for_func=False) == (True, 0)
assert checkodesol(f(x).diff(x, 2), [Eq(f(x), C1 + C2*x),
Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)]) == \
[(True, 0), (True, 0), (False, C2)]
assert checkodesol(f(x).diff(x, 2), set([Eq(f(x), C1 + C2*x),
Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)])) == \
set([(True, 0), (True, 0), (False, C2)])
assert checkodesol(f(x).diff(x) - 1/f(x)/2, Eq(f(x)**2, x)) == \
[(True, 0), (True, 0)]
assert checkodesol(f(x).diff(x) - f(x), Eq(C1 * exp(x), f(x))) == (True, 0)
# Based on test_1st_homogeneous_coeff_ode2_eq3sol. Make sure that
# checkodesol tries back substituting f(x) when it can.
eq3 = x * exp(f(x) / x) + f(x) - x * f(x).diff(x)
sol3 = Eq(f(x), log(log(C1 / x)**(-x)))
assert not checkodesol(eq3, sol3)[1].has(f(x))
# This case was failing intermittently depending on hash-seed:
eqn = Eq(Derivative(x * Derivative(f(x), x), x) / x, exp(x))
sol = Eq(f(x), C1 + C2 * log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
eq = x**2 * (f(x).diff(x, 2)) + x * (f(x).diff(x)) + (2 * x**2 + 25) * f(x)
sol = Eq(
f(x),
C1 * besselj(5 * I,
sqrt(2) * x) + C2 * bessely(5 * I,
sqrt(2) * x))
assert checkodesol(eq, sol) == (True, 0)
eqs = [Eq(f(x).diff(x), f(x) + g(x)), Eq(g(x).diff(x), f(x) + g(x))]
sol = [Eq(f(x), -C1 + C2 * exp(2 * x)), Eq(g(x), C1 + C2 * exp(2 * x))]
assert checkodesol(eqs, sol) == (True, [0, 0])
def _print_jn(self, expr):
from sympy.functions import sqrt, besselj
x = expr.argument
expr2 = sqrt(S.Pi/(2*x))*besselj(expr.order + S.Half, x)
return self._print(expr2)
def _print_jn(self, expr):
from sympy.functions import sqrt, besselj
x = expr.argument
expr2 = sqrt(S.Pi/(2*x))*besselj(expr.order + S.Half, x)
return self._print(expr2)
def test_factorable():
eq = f(x) + f(x) * f(x).diff(x)
sols = [Eq(f(x), C1 - x), Eq(f(x), 0)]
assert set(sols) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sols) == 2 * [(True, 0)]
eq = f(x) * (f(x).diff(x) + f(x) * x + 2)
sols = [
Eq(f(x),
(C1 - sqrt(2) * sqrt(pi) * erfi(sqrt(2) * x / 2)) * exp(-x**2 / 2)),
Eq(f(x), 0)
]
assert set(sols) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sols) == 2 * [(True, 0)]
eq = (f(x).diff(x) + f(x) * x**2) * (f(x).diff(x, 2) + x * f(x))
sols = [
Eq(f(x),
C1 * airyai(-x) + C2 * airybi(-x)),
Eq(f(x), C1 * exp(-x**3 / 3))
]
assert set(sols) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sols[1]) == (True, 0)
eq = (f(x).diff(x) + f(x) * x**2) * (f(x).diff(x, 2) + f(x))
sols = [Eq(f(x), C1 * exp(-x**3 / 3)), Eq(f(x), C1 * sin(x) + C2 * cos(x))]
assert set(sols) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sols) == 2 * [(True, 0)]
eq = (f(x).diff(x)**2 - 1) * (f(x).diff(x)**2 - 4)
sols = [
Eq(f(x), C1 - x),
Eq(f(x), C1 + x),
Eq(f(x), C1 + 2 * x),
Eq(f(x), C1 - 2 * x)
]
assert set(sols) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sols) == 4 * [(True, 0)]
eq = (f(x).diff(x, 2) - exp(f(x))) * f(x).diff(x)
sol = Eq(f(x), C1)
assert sol == dsolve(eq, f(x), hint='factorable')
assert checkodesol(eq, sol) == (True, 0)
eq = (f(x).diff(x)**2 - 1) * (f(x) * f(x).diff(x) - 1)
sol = [
Eq(f(x), C1 - x),
Eq(f(x), -sqrt(C1 + 2 * x)),
Eq(f(x), sqrt(C1 + 2 * x)),
Eq(f(x), C1 + x)
]
assert set(sol) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sol) == 4 * [(True, 0)]
eq = Derivative(f(x), x)**4 - 2 * Derivative(f(x), x)**2 + 1
sol = [Eq(f(x), C1 - x), Eq(f(x), C1 + x)]
assert set(sol) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sol) == 2 * [(True, 0)]
eq = f(x)**2 * Derivative(f(x), x)**6 - 2 * f(x)**2 * Derivative(
f(x), x)**4 + f(x)**2 * Derivative(f(x), x)**2 - 2 * f(x) * Derivative(
f(x), x)**5 + 4 * f(x) * Derivative(
f(x), x)**3 - 2 * f(x) * Derivative(f(x), x) + Derivative(
f(x), x)**4 - 2 * Derivative(f(x), x)**2 + 1
sol = [
Eq(f(x), C1 - x),
Eq(f(x), -sqrt(C1 + 2 * x)),
Eq(f(x), sqrt(C1 + 2 * x)),
Eq(f(x), C1 + x)
]
assert set(sol) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sol) == 4 * [(True, 0)]
eq = (f(x).diff(x, 2) - exp(f(x))) * (f(x).diff(x, 2) + exp(f(x)))
raises(NotImplementedError, lambda: dsolve(eq, hint='factorable'))
eq = x**4 * f(x)**2 + 2 * x**4 * f(x) * Derivative(
f(x), (x, 2)
) + x**4 * Derivative(f(x), (x, 2))**2 + 2 * x**3 * f(x) * Derivative(
f(x), x) + 2 * x**3 * Derivative(f(x), x) * Derivative(
f(x), (x, 2)) - 7 * x**2 * f(x)**2 - 7 * x**2 * f(x) * Derivative(
f(x), (x, 2)) + x**2 * Derivative(f(x), x)**2 - 7 * x * f(
x) * Derivative(f(x), x) + 12 * f(x)**2
sol = [
Eq(f(x),
C1 * besselj(2, x) + C2 * bessely(2, x)),
Eq(f(x),
C1 * besselj(sqrt(3), x) + C2 * bessely(sqrt(3), x))
]
assert set(sol) == set(dsolve(eq, f(x), hint='factorable'))
assert checkodesol(eq, sol) == 2 * [(True, 0)]
def _get_examples_ode_sol_factorable():
""" some hints are marked as xfail for examples because they missed additional algebraic solution
which could be found by Factorable hint. Fact_01 raise exception for
nth_linear_constant_coeff_undetermined_coefficients"""
return {
'hint': "factorable",
'func': f(x),
'examples': {
'fact_01': {
'eq':
f(x) + f(x) * f(x).diff(x),
'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)],
'XFAIL': [
'separable', '1st_exact', '1st_linear', 'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep', 'lie_group',
'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
'nth_linear_constant_coeff_undetermined_coefficients'
]
},
'fact_02': {
'eq':
f(x) * (f(x).diff(x) + f(x) * x + 2),
'sol': [
Eq(f(x),
(C1 - sqrt(2) * sqrt(pi) * erfi(sqrt(2) * x / 2)) *
exp(-x**2 / 2)),
Eq(f(x), 0)
],
'XFAIL': ['Bernoulli', '1st_linear', 'lie_group']
},
'fact_03': {
'eq':
(f(x).diff(x) + f(x) * x**2) * (f(x).diff(x, 2) + x * f(x)),
'sol': [
Eq(f(x),
C1 * airyai(-x) + C2 * airybi(-x)),
Eq(f(x), C1 * exp(-x**3 / 3))
]
},
'fact_04': {
'eq': (f(x).diff(x) + f(x) * x**2) * (f(x).diff(x, 2) + f(x)),
'sol': [
Eq(f(x), C1 * exp(-x**3 / 3)),
Eq(f(x),
C1 * sin(x) + C2 * cos(x))
]
},
'fact_05': {
'eq': (f(x).diff(x)**2 - 1) * (f(x).diff(x)**2 - 4),
'sol': [
Eq(f(x), C1 - x),
Eq(f(x), C1 + x),
Eq(f(x), C1 + 2 * x),
Eq(f(x), C1 - 2 * x)
]
},
'fact_06': {
'eq': (f(x).diff(x, 2) - exp(f(x))) * f(x).diff(x),
'sol': [Eq(f(x), C1)]
},
'fact_07': {
'eq': (f(x).diff(x)**2 - 1) * (f(x) * f(x).diff(x) - 1),
'sol': [
Eq(f(x), C1 - x),
Eq(f(x), -sqrt(C1 + 2 * x)),
Eq(f(x), sqrt(C1 + 2 * x)),
Eq(f(x), C1 + x)
]
},
'fact_08': {
'eq': Derivative(f(x), x)**4 - 2 * Derivative(f(x), x)**2 + 1,
'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x)]
},
'fact_09': {
'eq':
f(x)**2 * Derivative(f(x), x)**6 -
2 * f(x)**2 * Derivative(f(x), x)**4 +
f(x)**2 * Derivative(f(x), x)**2 -
2 * f(x) * Derivative(f(x), x)**5 +
4 * f(x) * Derivative(f(x), x)**3 -
2 * f(x) * Derivative(f(x), x) + Derivative(f(x), x)**4 -
2 * Derivative(f(x), x)**2 + 1,
'sol': [
Eq(f(x), C1 - x),
Eq(f(x), -sqrt(C1 + 2 * x)),
Eq(f(x), sqrt(C1 + 2 * x)),
Eq(f(x), C1 + x)
]
},
'fact_10': {
'eq':
x**4 * f(x)**2 + 2 * x**4 * f(x) * Derivative(f(x), (x, 2)) +
x**4 * Derivative(f(x), (x, 2))**2 +
2 * x**3 * f(x) * Derivative(f(x), x) +
2 * x**3 * Derivative(f(x), x) * Derivative(f(x), (x, 2)) -
7 * x**2 * f(x)**2 -
7 * x**2 * f(x) * Derivative(f(x), (x, 2)) +
x**2 * Derivative(f(x), x)**2 -
7 * x * f(x) * Derivative(f(x), x) + 12 * f(x)**2,
'sol': [
Eq(f(x),
C1 * besselj(2, x) + C2 * bessely(2, x)),
Eq(f(x),
C1 * besselj(sqrt(3), x) + C2 * bessely(sqrt(3), x))
]
},
'fact_11': {
'eq':
(f(x).diff(x, 2) - exp(f(x))) * (f(x).diff(x, 2) + exp(f(x))),
'sol':
[], #currently dsolve doesn't return any solution for this example
'XFAIL': ['factorable']
},
}
}
def _get_examples_ode_sol_factorable():
""" some hints are marked as xfail for examples because they missed additional algebraic solution
which could be found by Factorable hint. Fact_01 raise exception for
nth_linear_constant_coeff_undetermined_coefficients"""
y = Dummy('y')
return {
'hint': "factorable",
'func': f(x),
'examples':{
'fact_01': {
'eq': f(x) + f(x)*f(x).diff(x),
'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)],
'XFAIL': ['separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep', '1st_homogeneous_coeff_subs_dep_div_indep',
'lie_group', 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
'nth_linear_constant_coeff_undetermined_coefficients']
},
'fact_02': {
'eq': f(x)*(f(x).diff(x)+f(x)*x+2),
'sol': [Eq(f(x), (C1 - sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2))*exp(-x**2/2)), Eq(f(x), 0)],
'XFAIL': ['Bernoulli', '1st_linear', 'lie_group']
},
'fact_03': {
'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + x*f(x)),
'sol': [Eq(f(x), C1*airyai(-x) + C2*airybi(-x)),Eq(f(x), C1*exp(-x**3/3))]
},
'fact_04': {
'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + f(x)),
'sol': [Eq(f(x), C1*exp(-x**3/3)), Eq(f(x), C1*sin(x) + C2*cos(x))]
},
'fact_05': {
'eq': (f(x).diff(x)**2-1)*(f(x).diff(x)**2-4),
'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x), Eq(f(x), C1 + 2*x), Eq(f(x), C1 - 2*x)]
},
'fact_06': {
'eq': (f(x).diff(x, 2)-exp(f(x)))*f(x).diff(x),
'sol': [Eq(f(x), C1)]
},
'fact_07': {
'eq': (f(x).diff(x)**2-1)*(f(x)*f(x).diff(x)-1),
'sol': [Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)),Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x)]
},
'fact_08': {
'eq': Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1,
'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x)]
},
'fact_09': {
'eq': f(x)**2*Derivative(f(x), x)**6 - 2*f(x)**2*Derivative(f(x),
x)**4 + f(x)**2*Derivative(f(x), x)**2 - 2*f(x)*Derivative(f(x),
x)**5 + 4*f(x)*Derivative(f(x), x)**3 - 2*f(x)*Derivative(f(x),
x) + Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1,
'sol': [Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)),
Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x)]
},
'fact_10': {
'eq': x**4*f(x)**2 + 2*x**4*f(x)*Derivative(f(x), (x, 2)) + x**4*Derivative(f(x),
(x, 2))**2 + 2*x**3*f(x)*Derivative(f(x), x) + 2*x**3*Derivative(f(x),
x)*Derivative(f(x), (x, 2)) - 7*x**2*f(x)**2 - 7*x**2*f(x)*Derivative(f(x),
(x, 2)) + x**2*Derivative(f(x), x)**2 - 7*x*f(x)*Derivative(f(x), x) + 12*f(x)**2,
'sol': [Eq(f(x), C1*besselj(2, x) + C2*bessely(2, x)), Eq(f(x), C1*besselj(sqrt(3),
x) + C2*bessely(sqrt(3), x))]
},
'fact_11': {
'eq': (f(x).diff(x, 2)-exp(f(x)))*(f(x).diff(x, 2)+exp(f(x))),
'sol': [], #currently dsolve doesn't return any solution for this example
'XFAIL': ['factorable']
},
#Below examples were added for the issue: https://github.com/sympy/sympy/issues/15889
'fact_12': {
'eq': exp(f(x).diff(x))-f(x)**2,
'sol': [Eq(NonElementaryIntegral(1/log(y**2), (y, f(x))), C1 + x)],
'XFAIL': ['lie_group'] #It shows not implemented error for lie_group.
},
'fact_13': {
'eq': f(x).diff(x)**2 - f(x)**3,
'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))],
'XFAIL': ['lie_group'] #It shows not implemented error for lie_group.
},
'fact_14': {
'eq': f(x).diff(x)**2 - f(x),
'sol': [Eq(f(x), C1**2/4 - C1*x/2 + x**2/4)]
},
'fact_15': {
'eq': f(x).diff(x)**2 - f(x)**2,
'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1*exp(-x))]
},
'fact_16': {
'eq': f(x).diff(x)**2 - f(x)**3,
'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))]
},
}
}
