def test_issue_15889():
eq = exp(f(x).diff(x)) - f(x)**2
sol = Eq(NonElementaryIntegral(1 / log(y**2), (y, f(x))), C1 + x)
assert sol.dummy_eq(dsolve(eq))
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x)**2 - f(x)**3
sol = Eq(f(x), 4 / (C1**2 - 2 * C1 * x + x**2))
assert sol == dsolve(eq)
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x)**2 - f(x)
sol = Eq(f(x), C1**2 / 4 - C1 * x / 2 + x**2 / 4)
assert sol == dsolve(eq)
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x)**2 - f(x)**2
sol = [Eq(f(x), C1 * exp(x)), Eq(f(x), C1 * exp(-x))]
assert sol == dsolve(eq)
assert checkodesol(eq, sol) == 2 * [(True, 0)]
eq = f(x).diff(x)**2 - f(x)**3
sol = Eq(f(x), 4 / (C1**2 - 2 * C1 * x + x**2))
assert sol == dsolve(eq)
assert checkodesol(eq, sol) == (True, 0)
def test_issue_15913():
eq = -C1/x - 2*x*f(x) - f(x) + Derivative(f(x), x)
sol = C2*exp(x**2 + x) + exp(x**2 + x)*Integral(C1*exp(-x**2 - x)/x, x)
assert checkodesol(eq, sol) == (True, 0)
sol = C1 + C2*exp(-x*y)
eq = Derivative(y*f(x), x) + f(x).diff(x, 2)
assert checkodesol(eq, sol, f(x)) == (True, 0)
def test_nth_algebraic_redundant_solutions():
# This one has a redundant solution that should be removed
eqn = f(x) * f(x).diff(x)
soln = Eq(f(x), C1)
assert checkodesol(eqn, soln, order=1, solve_for_func=False)[0]
assert soln == dsolve(eqn, f(x), hint='nth_algebraic')
assert soln == dsolve(eqn, f(x))
# This has two integral solutions and no algebraic solutions
eqn = (diff(f(x)) - x) * (diff(f(x)) + x)
sol = [Eq(f(x), C1 - x**2 / 2), Eq(f(x), C1 + x**2 / 2)]
assert all(c[0]
for c in checkodesol(eqn, sol, order=1, solve_for_func=False))
assert set(sol) == set(dsolve(eqn, f(x), hint='nth_algebraic'))
assert set(sol) == set(dsolve(eqn, f(x)))
eqn = f(x) + f(x) * f(x).diff(x)
solns = [Eq(f(x), 0), Eq(f(x), C1 - x)]
assert all(c[0]
for c in checkodesol(eqn, solns, order=1, solve_for_func=False))
assert set(solns) == set(dsolve(eqn, f(x)))
solns = [Eq(f(x), exp(x)), Eq(f(x), C1 * exp(C2 * x))]
solns_final = _remove_redundant_solutions(eqn, solns, 2, x)
assert solns_final == [Eq(f(x), C1 * exp(C2 * x))]
# This one needs a substitution f' = g.
eqn = -exp(x) + (x * Derivative(f(x), (x, 2)) + Derivative(f(x), x)) / x
sol = Eq(f(x), C1 + C2 * log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x))
def test_2nd_power_series_regular():
C1, C2 = symbols("C1 C2")
eq = x**2 * (f(x).diff(x, 2)) - 3 * x * (f(x).diff(x)) + (4 * x + 4) * f(x)
sol = Eq(f(x),
C1 * x**2 * (-16 * x**3 / 9 + 4 * x**2 - 4 * x + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = 4 * x**2 * (f(x).diff(
x, 2)) - 8 * x**2 * (f(x).diff(x)) + (4 * x**2 + 1) * f(x)
sol = Eq(
f(x),
C1 * sqrt(x) * (x**4 / 24 + x**3 / 6 + x**2 / 2 + x + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = x**2 * (f(x).diff(x, 2)) - x**2 * (f(x).diff(x)) + (x**2 - 2) * f(x)
sol = Eq(
f(x),
C1 *
(-x**6 / 720 - 3 * x**5 / 80 - x**4 / 8 + x**2 / 2 + x / 2 + 1) / x +
C2 * x**2 * (-x**3 / 60 + x**2 / 20 + x / 2 + 1) + O(x**6))
assert dsolve(eq) == sol
assert checkodesol(eq, sol) == (True, 0)
eq = x**2 * (f(x).diff(
x, 2)) + x * (f(x).diff(x)) + (x**2 - Rational(1, 4)) * f(x)
sol = Eq(
f(x),
C1 * (x**4 / 24 - x**2 / 2 + 1) / sqrt(x) + C2 * sqrt(x) *
(x**4 / 120 - x**2 / 6 + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
def test_2nd_power_series_ordinary():
C1, C2 = symbols("C1 C2")
eq = f(x).diff(x, 2) - x * f(x)
assert classify_ode(eq) == ('2nd_linear_airy', '2nd_power_series_ordinary')
sol = Eq(f(x), C2 * (x**3 / 6 + 1) + C1 * x * (x**3 / 12 + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_ordinary') == sol
assert checkodesol(eq, sol) == (True, 0)
sol = Eq(
f(x),
C2 * ((x + 2)**4 / 6 + (x + 2)**3 / 6 - (x + 2)**2 + 1) + C1 *
(x + (x + 2)**4 / 12 - (x + 2)**3 / 3 + S(2)) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_ordinary', x0=-2) == sol
# FIXME: Solution should be O((x+2)**6)
# assert checkodesol(eq, sol) == (True, 0)
sol = Eq(f(x), C2 * x + C1 + O(x**2))
assert dsolve(eq, hint='2nd_power_series_ordinary', n=2) == sol
assert checkodesol(eq, sol) == (True, 0)
eq = (1 + x**2) * (f(x).diff(x, 2)) + 2 * x * (f(x).diff(x)) - 2 * f(x)
assert classify_ode(eq) == ('2nd_hypergeometric',
'2nd_hypergeometric_Integral',
'2nd_power_series_ordinary')
sol = Eq(f(x), C2 * (-x**4 / 3 + x**2 + 1) + C1 * x + O(x**6))
assert dsolve(eq, hint='2nd_power_series_ordinary') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + x * (f(x).diff(x)) + f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary', )
sol = Eq(
f(x),
C2 * (x**4 / 8 - x**2 / 2 + 1) + C1 * x * (-x**2 / 3 + 1) + O(x**6))
assert dsolve(eq) == sol
# FIXME: checkodesol fails for this solution...
# assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + f(x).diff(x) - x * f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary', )
sol = Eq(
f(x),
C2 * (-x**4 / 24 + x**3 / 6 + 1) + C1 * x *
(x**3 / 24 + x**2 / 6 - x / 2 + 1) + O(x**6))
assert dsolve(eq) == sol
# FIXME: checkodesol fails for this solution...
# assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + x * f(x)
assert classify_ode(eq) == ('2nd_linear_airy', '2nd_power_series_ordinary')
sol = Eq(
f(x),
C2 * (x**6 / 180 - x**3 / 6 + 1) + C1 * x * (-x**3 / 12 + 1) + O(x**7))
assert dsolve(eq, hint='2nd_power_series_ordinary', n=7) == sol
assert checkodesol(eq, sol) == (True, 0)
def test_nth_algebraic():
eqn = Eq(Derivative(f(x), x), Derivative(g(x), x))
sol = Eq(f(x), C1 + g(x))
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x),
hint='nth_algebraic'), dsolve(eqn,
f(x),
hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = (diff(f(x)) - x) * (diff(f(x)) + x)
sol = [Eq(f(x), C1 - x**2 / 2), Eq(f(x), C1 + x**2 / 2)]
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert set(sol) == set(dsolve(eqn, f(x), hint='nth_algebraic'))
assert set(sol) == set(dsolve(eqn, f(x)))
eqn = (1 - sin(f(x))) * f(x).diff(x)
sol = Eq(f(x), C1)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
M, m, r, t = symbols('M m r t')
phi = Function('phi')
eqn = Eq(-M * phi(t).diff(t),
Rational(3, 2) * m * r**2 * phi(t).diff(t) * phi(t).diff(t, t))
solns = [
Eq(phi(t), C1),
Eq(phi(t), C1 + C2 * t - M * t**2 / (3 * m * r**2))
]
assert checkodesol(eqn, solns[0], order=2, solve_for_func=False)[0]
assert checkodesol(eqn, solns[1], order=2, solve_for_func=False)[0]
assert set(solns) == set(dsolve(eqn, phi(t), hint='nth_algebraic'))
assert set(solns) == set(dsolve(eqn, phi(t)))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x)
sol = Eq(f(x), C1 + C2 * x)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1)
sol = Eq(f(x), C1 + C2 * x)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = f(x) * f(x).diff(x) * f(x).diff(x,
x) * (f(x) - 1) * (f(x).diff(x) - x)
solns = [Eq(f(x), C1 + x**2 / 2), Eq(f(x), C1 + C2 * x)]
assert checkodesol(eqn, solns[0], order=2, solve_for_func=False)[0]
assert checkodesol(eqn, solns[1], order=2, solve_for_func=False)[0]
assert set(solns) == set(dsolve(eqn, f(x), hint='nth_algebraic'))
assert set(solns) == set(dsolve(eqn, f(x)))
def test_series():
C1 = Symbol("C1")
eq = f(x).diff(x) - f(x)
sol = Eq(f(x), C1 + C1*x + C1*x**2/2 + C1*x**3/6 + C1*x**4/24 +
C1*x**5/120 + O(x**6))
assert dsolve(eq, hint='1st_power_series') == sol
assert checkodesol(eq, sol, order=1)[0]
eq = f(x).diff(x) - x*f(x)
sol = Eq(f(x), C1*x**4/8 + C1*x**2/2 + C1 + O(x**6))
assert dsolve(eq, hint='1st_power_series') == sol
assert checkodesol(eq, sol, order=1)[0]
eq = f(x).diff(x) - sin(x*f(x))
sol = Eq(f(x), (x - 2)**2*(1+ sin(4))*cos(4) + (x - 2)*sin(4) + 2 + O(x**3))
assert dsolve(eq, hint='1st_power_series', ics={f(2): 2}, n=3) == sol
def test_user_infinitesimals():
x = Symbol("x") # assuming x is real generates an error
eq = x * (f(x).diff(x)) + 1 - f(x)**2
sol = Eq(f(x), (C1 + x**2) / (C1 - x**2))
infinitesimals = {'xi': sqrt(f(x) - 1) / sqrt(f(x) + 1), 'eta': 0}
assert dsolve(eq, hint='lie_group', **infinitesimals) == sol
assert checkodesol(eq, sol) == (True, 0)
def test_Bernoulli():
# Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n
eq = Eq(x*f(x).diff(x) + f(x) - f(x)**2, 0)
sol = dsolve(eq, f(x), hint='Bernoulli')
assert sol == Eq(f(x), 1/(C1*x + 1))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = f(x).diff(x) - y*f(x)
sol = dsolve(eq, hint='Bernoulli')
assert sol == Eq(f(x), C1*exp(x*y))
assert checkodesol(eq, sol)[0]
eq = f(x)*f(x).diff(x) - 1
sol = dsolve(eq,hint='Bernoulli')
assert sol == [Eq(f(x), -sqrt(C1 + 2*x)), Eq(f(x), sqrt(C1 + 2*x))]
assert checkodesol(eq, sol) == [(True, 0), (True, 0)]
def test_eval(kamkenumber):
print("kamkenumber = ",kamkenumber)
odesol = dsolve(kamke1_1[kamkenumber],y)
print("odesol = ",odesol)
#expected = solution_kamke1[kamkenumber]
#print("odeexpected = ",expected)
#assert odesol == expected
assert checkodesol(kamke1_1[kamkenumber],odesol)[0]
def test_issue_13060():
A, B = symbols("A B", cls=Function)
t = Symbol("t")
eq = [
Eq(Derivative(A(t), t),
A(t) * B(t)),
Eq(Derivative(B(t), t),
A(t) * B(t))
]
sol = dsolve(eq)
assert checkodesol(eq, sol) == (True, [0, 0])
def find_riccati_ode(ratfunc, x, yf):
y = ratfunc
yp = y.diff(x)
q1 = rand_rational_function(x, 1, 3)
q2 = rand_rational_function(x, 1, 3)
while q2 == 0:
q2 = rand_rational_function(x, 1, 3)
q0 = ratsimp(yp - q1 * y - q2 * y**2)
eq = Eq(yf.diff(), q0 + q1 * yf + q2 * yf**2)
sol = Eq(yf, y)
assert checkodesol(eq, sol) == (True, 0)
return eq, q0, q1, q2
def _test_all_examples_for_one_hint(our_hint, all_examples=[], runxfail=None):
if all_examples == []:
all_examples = _get_all_examples()
match_list, unsolve_list, exception_list = [], [], []
for ode_example in all_examples:
eq = ode_example['eq']
expected_sol = ode_example['sol']
example = ode_example['example_name']
xfail = our_hint in ode_example['XFAIL']
xpass = True
if runxfail and not xfail:
continue
if our_hint in classify_ode(eq):
match_list.append(example)
try:
dsolve_sol = dsolve(eq, hint=our_hint)
expected_checkodesol = [(True, 0)
for i in range(len(expected_sol))]
if len(expected_sol) == 1:
expected_checkodesol = (True, 0)
if checkodesol(eq, dsolve_sol) != expected_checkodesol:
unsolve_list.append(example)
message = dsol_incorrect_msg.format(hint=our_hint,
eq=eq,
sol=expected_sol,
dsolve_sol=dsolve_sol)
if runxfail is not None:
raise AssertionError(message)
except Exception as e:
exception_list.append(example)
if runxfail is not None:
print(
exception_msg.format(e=str(e),
hint=our_hint,
example=example,
eq=eq))
traceback.print_exc()
xpass = False
if xpass and xfail:
print(example, "is now passing for the hint", our_hint)
if runxfail is None:
match_count = len(match_list)
solved = len(match_list) - len(unsolve_list) - len(exception_list)
msg = check_hint_msg.format(hint=our_hint,
matched=match_count,
solve=solved,
unsolve=unsolve_list,
exceptions=exception_list)
print(msg)
def check_dummy_sol(eq, solse, dummy_sym):
"""
Helper function to check if actual solution
matches expected solution if actual solution
contains dummy symbols.
"""
if isinstance(eq, Eq):
eq = eq.lhs - eq.rhs
_, funcs = match_riccati(eq, f, x)
sols = solve_riccati(f(x), x, *funcs)
C1 = Dummy('C1')
sols = [sol.subs(C1, dummy_sym) for sol in sols]
assert all([x[0] for x in checkodesol(eq, sols)])
assert all([s1.dummy_eq(s2, dummy_sym) for s1, s2 in zip(sols, solse)])
def _ode_solver_test(ode_examples):
our_hint = ode_examples['hint']
for example in ode_examples['examples']:
eq = ode_examples['examples'][example]['eq']
sol = ode_examples['examples'][example]['sol']
if our_hint not in classify_ode(eq):
message = hint_message.format(example=example,
eq=eq,
our_hint=our_hint)
raise AssertionError(message)
dsolve_sol = dsolve(eq, hint=our_hint)
if dsolve_sol not in sol:
message = expected_sol_message.format(example=example,
eq=eq,
sol=sol,
dsolve_sol=dsolve_sol)
raise AssertionError(message)
expected_checkodesol = [(True, 0) for i in range(len(sol))]
if checkodesol(eq, sol) != expected_checkodesol:
message = checkodesol.format(example=example, eq=eq)
raise AssertionError(message)
def test_2nd_power_series_regular():
C1, C2, a = symbols("C1 C2 a")
eq = x**2 * (f(x).diff(x, 2)) - 3 * x * (f(x).diff(x)) + (4 * x + 4) * f(x)
sol = Eq(f(x),
C1 * x**2 * (-16 * x**3 / 9 + 4 * x**2 - 4 * x + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = 4 * x**2 * (f(x).diff(
x, 2)) - 8 * x**2 * (f(x).diff(x)) + (4 * x**2 + 1) * f(x)
sol = Eq(
f(x),
C1 * sqrt(x) * (x**4 / 24 + x**3 / 6 + x**2 / 2 + x + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = x**2 * (f(x).diff(x, 2)) - x**2 * (f(x).diff(x)) + (x**2 - 2) * f(x)
sol = Eq(
f(x),
C1 *
(-x**6 / 720 - 3 * x**5 / 80 - x**4 / 8 + x**2 / 2 + x / 2 + 1) / x +
C2 * x**2 * (-x**3 / 60 + x**2 / 20 + x / 2 + 1) + O(x**6))
assert dsolve(eq) == sol
assert checkodesol(eq, sol) == (True, 0)
eq = x**2 * (f(x).diff(
x, 2)) + x * (f(x).diff(x)) + (x**2 - Rational(1, 4)) * f(x)
sol = Eq(
f(x),
C1 * (x**4 / 24 - x**2 / 2 + 1) / sqrt(x) + C2 * sqrt(x) *
(x**4 / 120 - x**2 / 6 + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = x * f(x).diff(x, 2) + f(x).diff(x) - a * x * f(x)
sol = Eq(f(x), C1 * (a**2 * x**4 / 64 + a * x**2 / 4 + 1) + O(x**6))
assert dsolve(eq, f(x), hint="2nd_power_series_regular") == sol
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + ((1 - x) / x) * f(x).diff(x) + (a / x) * f(x)
sol = Eq(f(x), C1*(-a*x**5*(a - 4)*(a - 3)*(a - 2)*(a - 1)/14400 + \
a*x**4*(a - 3)*(a - 2)*(a - 1)/576 - a*x**3*(a - 2)*(a - 1)/36 + \
a*x**2*(a - 1)/4 - a*x + 1) + O(x**6))
assert dsolve(eq, f(x), hint="2nd_power_series_regular") == sol
assert checkodesol(eq, sol) == (True, 0)
def test_nth_algebraic_prep2():
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]
assert sol == dsolve(eqn, f(x), prep=True, hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
def test_nth_algebraic_prep1():
eqn = Derivative(x * f(x), x, x, x)
sol = Eq(f(x), (C1 + C2 * x + C3 * x**2) / x)
assert checkodesol(eqn, sol, order=3, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=True, hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
def test_nth_algebraic_issue15999():
eqn = f(x).diff(x) - C1
sol = Eq(f(x), C1 * x + C2) # Correct solution
assert checkodesol(eqn, sol, order=1, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x), hint='nth_algebraic') == sol
assert dsolve(eqn, f(x)) == sol
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 _test_particular_example(our_hint,
ode_example=None,
solver_flag=False,
example_name=None):
if example_name is not None:
all_examples = _get_all_examples()
for example in all_examples:
if example['example_name'] == example_name:
eq = example['eq']
dsolve(eq, hint=our_hint)
else:
eq = ode_example['eq']
expected_sol = ode_example['sol']
example = ode_example['example_name']
xfail = our_hint in ode_example['XFAIL']
func = ode_example['func']
result = {'msg': '', 'xpass_msg': ''}
xpass = True
if solver_flag:
if our_hint not in classify_ode(eq, func):
message = hint_message.format(example=example,
eq=eq,
our_hint=our_hint)
raise AssertionError(message)
if our_hint in classify_ode(eq, func):
result['match_list'] = example
try:
dsolve_sol = dsolve(eq, func, hint=our_hint)
if solver_flag:
expect_sol_check = False
if type(dsolve_sol) == list:
if any(sub_sol not in dsolve_sol
for sub_sol in expected_sol):
expect_sol_check = True
else:
expect_sol_check = dsolve_sol not in expected_sol
if expect_sol_check:
message = expected_sol_message.format(
example=example,
eq=eq,
sol=expected_sol,
dsolve_sol=dsolve_sol)
raise AssertionError(message)
expected_checkodesol = [(True, 0)
for i in range(len(expected_sol))]
if len(expected_sol) == 1:
expected_checkodesol = (True, 0)
if checkodesol(eq, dsolve_sol) != expected_checkodesol:
result['unsolve_list'] = example
xpass = False
message = dsol_incorrect_msg.format(hint=our_hint,
eq=eq,
sol=expected_sol,
dsolve_sol=dsolve_sol)
if solver_flag:
message = checkodesol.format(example=example, eq=eq)
raise AssertionError(message)
else:
result['msg'] = 'AssertionError: ' + message
except Exception as e:
result['exception_list'] = example
if not solver_flag:
traceback.print_exc()
result['msg'] = exception_msg.format(e=str(e),
hint=our_hint,
example=example,
eq=eq)
xpass = False
if xpass and xfail:
result[
'xpass_msg'] = example + "is now passing for the hint" + our_hint
return result
def test_Riccati_special_minus2():
# Type: Riccati special alpha = -2, a*dy/dx + b*y**2 + c*y/x +d/x**2
eq = 2 * f(x).diff(x) + f(x)**2 - f(x) / x + 3 * x**(-2)
sol = dsolve(eq, f(x), hint='Riccati_special_minus2')
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_Bernoulli():
# Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n
eq = Eq(x * f(x).diff(x) + f(x) - f(x)**2, 0)
sol = dsolve(eq, f(x), hint='Bernoulli')
assert sol == Eq(f(x), 1 / (x * (C1 + 1 / x)))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
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 _test_particular_example(our_hint, ode_example, solver_flag=False):
eq = ode_example['eq']
expected_sol = ode_example['sol']
example = ode_example['example_name']
xfail = our_hint in ode_example['XFAIL']
func = ode_example['func']
result = {'msg': '', 'xpass_msg': ''}
checkodesol_XFAIL = ode_example['checkodesol_XFAIL']
xpass = True
if solver_flag:
if our_hint not in classify_ode(eq, func):
message = hint_message.format(example=example, eq=eq, our_hint=our_hint)
raise AssertionError(message)
if our_hint in classify_ode(eq, func):
result['match_list'] = example
try:
dsolve_sol = dsolve(eq, func, hint=our_hint)
except Exception as e:
dsolve_sol = []
result['exception_list'] = example
if not solver_flag:
traceback.print_exc()
result['msg'] = exception_msg.format(e=str(e), hint=our_hint, example=example, eq=eq)
xpass = False
if solver_flag and dsolve_sol!=[]:
expect_sol_check = False
if type(dsolve_sol)==list:
for sub_sol in expected_sol:
if sub_sol.has(Dummy):
expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol)
else:
expect_sol_check = sub_sol not in dsolve_sol
if expect_sol_check:
break
else:
expect_sol_check = dsolve_sol not in expected_sol
for sub_sol in expected_sol:
if sub_sol.has(Dummy):
expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol)
if expect_sol_check:
message = expected_sol_message.format(example=example, eq=eq, sol=expected_sol, dsolve_sol=dsolve_sol)
raise AssertionError(message)
expected_checkodesol = [(True, 0) for i in range(len(expected_sol))]
if len(expected_sol) == 1:
expected_checkodesol = (True, 0)
if not checkodesol_XFAIL:
if checkodesol(eq, dsolve_sol, solve_for_func=False) != expected_checkodesol:
result['unsolve_list'] = example
xpass = False
message = dsol_incorrect_msg.format(hint=our_hint, eq=eq, sol=expected_sol,dsolve_sol=dsolve_sol)
if solver_flag:
message = checkodesol_msg.format(example=example, eq=eq)
raise AssertionError(message)
else:
result['msg'] = 'AssertionError: ' + message
if xpass and xfail:
result['xpass_msg'] = example + "is now passing for the hint" + our_hint
return result