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_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_issue_4825():
raises(ValueError, lambda: dsolve(f(x, y).diff(x) - y * f(x, y), f(x)))
assert classify_ode(f(x, y).diff(x) - y*f(x, y), f(x), dict=True) == \
{'order': 0, 'default': None, 'ordered_hints': ()}
# See also issue 3793, test Z13.
raises(ValueError, lambda: dsolve(f(x).diff(x), f(y)))
assert classify_ode(f(x).diff(x), f(y), dict=True) == \
{'order': 0, 'default': None, 'ordered_hints': ()}
def test_dsolve_all_hint():
eq = f(x).diff(x)
output = dsolve(eq, hint='all')
# Match the Dummy variables:
sol1 = output['separable_Integral']
_y = sol1.lhs.args[1][0]
sol1 = output['1st_homogeneous_coeff_subs_dep_div_indep_Integral']
_u1 = sol1.rhs.args[1].args[1][0]
expected = {
'Bernoulli_Integral':
Eq(f(x), C1 + Integral(0, x)),
'1st_homogeneous_coeff_best':
Eq(f(x), C1),
'Bernoulli':
Eq(f(x), C1),
'nth_algebraic':
Eq(f(x), C1),
'nth_linear_euler_eq_homogeneous':
Eq(f(x), C1),
'nth_linear_constant_coeff_homogeneous':
Eq(f(x), C1),
'separable':
Eq(f(x), C1),
'1st_homogeneous_coeff_subs_indep_div_dep':
Eq(f(x), C1),
'nth_algebraic_Integral':
Eq(f(x), C1),
'1st_linear':
Eq(f(x), C1),
'1st_linear_Integral':
Eq(f(x), C1 + Integral(0, x)),
'lie_group':
Eq(f(x), C1),
'1st_homogeneous_coeff_subs_dep_div_indep':
Eq(f(x), C1),
'1st_homogeneous_coeff_subs_dep_div_indep_Integral':
Eq(log(x), C1 + Integral(-1 / _u1, (_u1, f(x) / x))),
'1st_power_series':
Eq(f(x), C1),
'separable_Integral':
Eq(Integral(1, (_y, f(x))), C1 + Integral(0, x)),
'1st_homogeneous_coeff_subs_indep_div_dep_Integral':
Eq(f(x), C1),
'best':
Eq(f(x), C1),
'best_hint':
'nth_algebraic',
'default':
'nth_algebraic',
'order':
1
}
assert output == expected
assert dsolve(eq, hint='best') == Eq(f(x), C1)
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)))
dsolve(eq1)
def test_dsolve_remove_redundant_solutions():
eq = (f(x) - 2) * f(x).diff(x)
sol = Eq(f(x), C1)
assert dsolve(eq) == sol
eq = (f(x) - sin(x)) * (f(x).diff(x, 2))
sol = {Eq(f(x), C1 + C2 * x), Eq(f(x), sin(x))}
assert set(dsolve(eq)) == sol
eq = (f(x)**2 - 2 * f(x) + 1) * f(x).diff(x, 3)
sol = Eq(f(x), C1 + C2 * x + C3 * x**2)
assert dsolve(eq) == sol
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_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_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_linear_3eq_order1_type4_skip():
if ON_TRAVIS:
skip("Too slow for travis.")
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)))
# sol1 = [Eq(x(t), (C1*exp(-2*Integral(t**3 + log(t), t)) + C2*(sqrt(3)*sin(sqrt(3)*Integral(t**3 + log(t), t))/6 \
# + cos(sqrt(3)*Integral(t**3 + log(t), t))/2) + C3*(-sin(sqrt(3)*Integral(t**3 + log(t), t))/2 \
# + sqrt(3)*cos(sqrt(3)*Integral(t**3 + log(t), t))/6))*exp(Integral(-t**2 - sin(t), t))),
# Eq(y(t), (C2*(sqrt(3)*sin(sqrt(3)*Integral(t**3 + log(t), t))/6 + cos(sqrt(3)* \
# Integral(t**3 + log(t), t))/2) + C3*(-sin(sqrt(3)*Integral(t**3 + log(t), t))/2 \
# + sqrt(3)*cos(sqrt(3)*Integral(t**3 + log(t), t))/6))*exp(Integral(-t**2 - sin(t), t))),
# Eq(z(t), (C1*exp(-2*Integral(t**3 + log(t), t)) + C2*cos(sqrt(3)*Integral(t**3 + log(t), t)) - \
# C3*sin(sqrt(3)*Integral(t**3 + log(t), t)))*exp(Integral(-t**2 - sin(t), t)))]
dsolve_sol = dsolve(eq1)
# dsolve_sol = [eq.subs(C3, -C3) for eq in dsolve_sol]
# assert all(simplify(s1.rhs - ds1.rhs) == 0 for s1, ds1 in zip(sol1, dsolve_sol))
assert checksysodesol(eq1, dsolve_sol) == (True, [0, 0, 0])
def func(i):
print("")
print("kamke number ",i)
# kamke ODEs start at 1 but a list start at 0
print("kamke ode = ",kamke1_1[i])
kamkesol = dsolve(kamke1_1[i],y)
print("kamkesol = ",kamkesol)
return(kamkesol)
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 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_issue_22604():
x1, x2 = symbols('x1, x2', cls = Function)
t, k1, k2, m1, m2 = symbols('t k1 k2 m1 m2', real = True)
k1, k2, m1, m2 = 1, 1, 1, 1
eq1 = Eq(m1*diff(x1(t), t, 2) + k1*x1(t) - k2*(x2(t) - x1(t)), 0)
eq2 = Eq(m2*diff(x2(t), t, 2) + k2*(x2(t) - x1(t)), 0)
eqs = [eq1, eq2]
[x1sol, x2sol] = dsolve(eqs, [x1(t), x2(t)], ics = {x1(0):0, x1(t).diff().subs(t,0):0, \
x2(0):1, x2(t).diff().subs(t,0):0})
assert x1sol == Eq(x1(t), sqrt(3 - sqrt(5))*(sqrt(10) + 5*sqrt(2))*cos(sqrt(2)*t*sqrt(3 - sqrt(5))/2)/20 + \
(-5*sqrt(2) + sqrt(10))*sqrt(sqrt(5) + 3)*cos(sqrt(2)*t*sqrt(sqrt(5) + 3)/2)/20)
assert x2sol == Eq(x2(t), (sqrt(5) + 5)*cos(sqrt(2)*t*sqrt(3 - sqrt(5))/2)/10 + (5 - sqrt(5))*cos(sqrt(2)*t*sqrt(sqrt(5) + 3)/2)/10)
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_issue_5770():
k = Symbol("k", real=True)
t = Symbol('t')
w = Function('w')
sol = dsolve(w(t).diff(t, 6) - k**6 * w(t), w(t))
assert len([s for s in sol.free_symbols if s.name.startswith('C')]) == 6
assert constantsimp((C1*cos(x) + C2*cos(x))*exp(x), {C1, C2}) == \
C1*cos(x)*exp(x)
assert constantsimp(C1*cos(x) + C2*cos(x) + C3*sin(x), {C1, C2, C3}) == \
C1*cos(x) + C3*sin(x)
assert constantsimp(exp(C1 + x), {C1}) == C1 * exp(x)
assert constantsimp(x + C1 + y, {C1, y}) == C1 + x
assert constantsimp(x + C1 + Integral(x, (x, 1, 2)), {C1}) == C1 + x
def calc(self):
f = Function('f')
de = diff(self.L, Derivative(f(t), t))
de = diff(de, t)
dw = diff(self.L, f(t))
eq = de - dw
eq = eq.subs({cos(f(t)): 1 - (((f(t))**2) / 2), sin(f(t)): f(t)})
print(eq)
eq = simplify(eq)
print(eq)
solution = dsolve(eq, f(t))
print(solution)
def calc(self): # The function that calculates the expression of f(t)
f = Function('f')
t = symbols('t')
#The lagrangian
self.L = ode.kinetic(self) - ode.potential(self)
print('the lagrangian is : ')
display(self.L)
##The different placebos for the Lagrange equation##
de = diff(self.L, Derivative(f(t), t))
de = diff(de, t)
dw = diff(self.L, f(t))
da = 0.5 * self.alpha * (self.v**2) #Dissipation function
da = da.subs({sin(f(t)): f(t)})
da = da.subs({cos(f(t)): 1 - (f(t))**2 / 2})
da = diff(da, Derivative(f(t), t))
eq = de - dw + da #LAgrange Equation
##############----------##############
##Solving the equation##
print('The equation is : ')
eq = simplify(eq)
display(eq)
solution = dsolve(eq, f(t))
print('The solution is : ')
display(solution)
#######----------#######
#Initial conditions:
cnd0 = Eq(solution.subs({t: 0}), self.x0)
cnd1 = Eq(solution.diff(t).subs({t: 0}), self.xd0)
display(cnd0)
#Solve for C1 and C2:
C1, C2 = symbols('C1 C2') #Temporary symbols
C1C2_sl = linsolve([cnd0, cnd1], (C1, C2))
#Substitute into the Solution of the equation
solution2 = simplify(solution.subs(C1C2_sl))
if 'C1' in str(solution2):
solution3 = solution2.subs({C1: 1})
if 'C2' in str(solution2):
solution3 = solution2.subs({C2: 2})
print('The solution is :')
display(solution3) #The Solution
return solution3
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 reduction_of_order(eq):
try:
return dsolve(eq)
except NotImplementedError:
atom_x = list(eq.atoms(Symbol))[0]
atom_f = list(eq.atoms(Function))[0]
if ode_order(eq, atom_f) > 1:
g = Function('g')(atom_x)
eq = eq.subs(atom_f.diff(atom_x), g)
if not (eq.has(atom_f)):
return dsolve(
reduction_of_order(eq).subs(g, atom_f.diff(atom_x)))
else:
raise NotImplementedError
else:
raise NotImplementedError
else:
raise NotImplementedError
def test_issue_22523():
N, s = symbols('N s')
rho = Function('rho')
# intentionally use 4.0 to confirm issue with nfloat
# works here
eqn = 4.0*N*sqrt(N - 1)*rho(s) + (4*s**2*(N - 1) + (N - 2*s*(N - 1))**2
)*Derivative(rho(s), (s, 2))
match = classify_ode(eqn, dict=True, hint='all')
assert match['2nd_power_series_ordinary']['terms'] == 5
C1, C2 = symbols('C1,C2')
sol = dsolve(eqn, hint='2nd_power_series_ordinary')
# there is no r(2.0) in this result
assert filldedent(sol) == filldedent(str('''
Eq(rho(s), C2*(1 - 4.0*s**4*sqrt(N - 1.0)/N + 0.666666666666667*s**4/N
- 2.66666666666667*s**3*sqrt(N - 1.0)/N - 2.0*s**2*sqrt(N - 1.0)/N +
9.33333333333333*s**4*sqrt(N - 1.0)/N**2 - 0.666666666666667*s**4/N**2
+ 2.66666666666667*s**3*sqrt(N - 1.0)/N**2 -
5.33333333333333*s**4*sqrt(N - 1.0)/N**3) + C1*s*(1.0 -
1.33333333333333*s**3*sqrt(N - 1.0)/N - 0.666666666666667*s**2*sqrt(N
- 1.0)/N + 1.33333333333333*s**3*sqrt(N - 1.0)/N**2) + O(s**6))'''))
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 _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_dsolve_ics():
# Maybe this should just use one of the solutions instead of raising...
with raises(NotImplementedError):
dsolve(f(x).diff(x) - sqrt(f(x)), ics={f(1): 1})
def pde_1st_linear_variable_coeff(eq, func, order, match, solvefun):
r"""
Solves a first order linear partial differential equation
with variable coefficients. The general form of this partial differential equation is
.. math:: a(x, y) \frac{df(x, y)}{dx} + a(x, y) \frac{df(x, y)}{dy}
+ c(x, y) f(x, y) - G(x, y)
where `a(x, y)`, `b(x, y)`, `c(x, y)` and `G(x, y)` are arbitrary functions
in `x` and `y`. This PDE is converted into an ODE by making the following transformation.
1] `\xi` as `x`
2] `\eta` as the constant in the solution to the differential equation
`\frac{dy}{dx} = -\frac{b}{a}`
Making the following substitutions reduces it to the linear ODE
.. math:: a(\xi, \eta)\frac{du}{d\xi} + c(\xi, \eta)u - d(\xi, \eta) = 0
which can be solved using dsolve.
The general form of this PDE is::
>>> from sympy.solvers.pde import pdsolve
>>> from sympy.abc import x, y
>>> from sympy import Function, pprint
>>> a, b, c, G, f= [Function(i) for i in ['a', 'b', 'c', 'G', 'f']]
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a(x, y)*u + b(x, y)*ux + c(x, y)*uy - G(x,y)
>>> pprint(genform)
d d
-G(x, y) + a(x, y)*f(x, y) + b(x, y)*--(f(x, y)) + c(x, y)*--(f(x, y))
dx dy
Examples
========
>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, diff, pprint, exp
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2
>>> pdsolve(eq)
f(x, y) == F(x*y)*exp(y**2/2) + 1
References
==========
- Viktor Grigoryan, "Partial Differential Equations"
Math 124A - Fall 2010, pp.7
"""
from sympy.integrals.integrals import integrate
from sympy.solvers.ode import dsolve
xi, eta = symbols("xi eta")
f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
e = -match[match['e']]
if not d:
# To deal with cases like b*ux = e or c*uy = e
if not (b and c):
if c:
try:
tsol = integrate(e/c, y)
except NotImplementedError:
raise NotImplementedError("Unable to find a solution"
" due to inability of integrate")
else:
return Eq(f(x,y), solvefun(x) + tsol)
if b:
try:
tsol = integrate(e/b, x)
except NotImplementedError:
raise NotImplementedError("Unable to find a solution"
" due to inability of integrate")
else:
return Eq(f(x,y), solvefun(y) + tsol)
if not c:
# To deal with cases when c is 0, a simpler method is used.
# The PDE reduces to b*(u.diff(x)) + d*u = e, which is a linear ODE in x
plode = f(x).diff(x)*b + d*f(x) - e
sol = dsolve(plode, f(x))
syms = sol.free_symbols - plode.free_symbols - set([x, y])
rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, y)
return Eq(f(x, y), rhs)
if not b:
# To deal with cases when b is 0, a simpler method is used.
# The PDE reduces to c*(u.diff(y)) + d*u = e, which is a linear ODE in y
plode = f(y).diff(y)*c + d*f(y) - e
sol = dsolve(plode, f(y))
syms = sol.free_symbols - plode.free_symbols - set([x, y])
rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, x)
return Eq(f(x, y), rhs)
dummy = Function('d')
h = (c/b).subs(y, dummy(x))
sol = dsolve(dummy(x).diff(x) - h, dummy(x))
if isinstance(sol, list):
sol = sol[0]
solsym = sol.free_symbols - h.free_symbols - set([x, y])
if len(solsym) == 1:
solsym = solsym.pop()
etat = (solve(sol, solsym)[0]).subs(dummy(x), y)
ysub = solve(eta - etat, y)[0]
deq = (b*(f(x).diff(x)) + d*f(x) - e).subs(y, ysub)
final = (dsolve(deq, f(x), hint='1st_linear')).rhs
if isinstance(final, list):
final = final[0]
finsyms = final.free_symbols - deq.free_symbols - set([x, y])
rhs = _simplify_variable_coeff(final, finsyms, solvefun, etat)
return Eq(f(x, y), rhs)
else:
raise NotImplementedError("Cannot solve the partial differential equation due"
" to inability of constantsimp")
def test_issue_5095():
f = Function('f')
raises(ValueError, lambda: dsolve(f(x).diff(x)**2, f(x), 'fdsjf'))
def test_dsolve_options():
eq = x * f(x).diff(x) + f(x)
a = dsolve(eq, hint='all')
b = dsolve(eq, hint='all', simplify=False)
c = dsolve(eq, hint='all_Integral')
keys = [
'1st_exact', '1st_exact_Integral', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear',
'1st_linear_Integral', 'Bernoulli', 'Bernoulli_Integral',
'almost_linear', 'almost_linear_Integral', 'best', 'best_hint',
'default', 'lie_group', 'nth_linear_euler_eq_homogeneous', 'order',
'separable', 'separable_Integral'
]
Integral_keys = [
'1st_exact_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_linear_Integral', 'Bernoulli_Integral', 'almost_linear_Integral',
'best', 'best_hint', 'default', 'nth_linear_euler_eq_homogeneous',
'order', 'separable_Integral'
]
assert sorted(a.keys()) == keys
assert a['order'] == ode_order(eq, f(x))
assert a['best'] == Eq(f(x), C1 / x)
assert dsolve(eq, hint='best') == Eq(f(x), C1 / x)
assert a['default'] == 'separable'
assert a['best_hint'] == 'separable'
assert not a['1st_exact'].has(Integral)
assert not a['separable'].has(Integral)
assert not a['1st_homogeneous_coeff_best'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not a['1st_linear'].has(Integral)
assert a['1st_linear_Integral'].has(Integral)
assert a['1st_exact_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert a['separable_Integral'].has(Integral)
assert sorted(b.keys()) == keys
assert b['order'] == ode_order(eq, f(x))
assert b['best'] == Eq(f(x), C1 / x)
assert dsolve(eq, hint='best', simplify=False) == Eq(f(x), C1 / x)
assert b['default'] == 'separable'
assert b['best_hint'] == '1st_linear'
assert a['separable'] != b['separable']
assert a['1st_homogeneous_coeff_subs_dep_div_indep'] != \
b['1st_homogeneous_coeff_subs_dep_div_indep']
assert a['1st_homogeneous_coeff_subs_indep_div_dep'] != \
b['1st_homogeneous_coeff_subs_indep_div_dep']
assert not b['1st_exact'].has(Integral)
assert not b['separable'].has(Integral)
assert not b['1st_homogeneous_coeff_best'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not b['1st_linear'].has(Integral)
assert b['1st_linear_Integral'].has(Integral)
assert b['1st_exact_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert b['separable_Integral'].has(Integral)
assert sorted(c.keys()) == Integral_keys
raises(ValueError, lambda: dsolve(eq, hint='notarealhint'))
raises(ValueError, lambda: dsolve(eq, hint='Liouville'))
assert dsolve(f(x).diff(x) - 1/f(x)**2, hint='all')['best'] == \
dsolve(f(x).diff(x) - 1/f(x)**2, hint='best')
assert dsolve(f(x) + f(x).diff(x) + sin(x).diff(x) + 1, f(x),
hint="1st_linear_Integral") == \
Eq(f(x), (C1 + Integral((-sin(x).diff(x) - 1)*
exp(Integral(1, x)), x))*exp(-Integral(1, x)))
def test_solve_ics():
# Basic tests that things work from dsolve.
assert dsolve(f(x).diff(x) - 1/f(x), f(x), ics={f(1): 2}) == \
Eq(f(x), sqrt(2 * x + 2))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(0): 1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(x).diff(x).subs(x, 0):
1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x, x) + f(x),
f(x),
ics={
f(0): 1,
f(x).diff(x).subs(x, 0): 1
}) == Eq(f(x),
sin(x) + cos(x))
assert dsolve([f(x).diff(x) - f(x) + g(x),
g(x).diff(x) - g(x) - f(x)], [f(x), g(x)],
ics={
f(0): 1,
g(0): 0
}) == [Eq(f(x),
exp(x) * cos(x)),
Eq(g(x),
exp(x) * sin(x))]
# Test cases where dsolve returns two solutions.
eq = (x**2 * f(x)**2 - x).diff(x)
assert dsolve(eq, f(x), ics={f(1): 0}) == [
Eq(f(x), -sqrt(x - 1) / x),
Eq(f(x),
sqrt(x - 1) / x)
]
assert dsolve(eq, f(x), ics={f(x).diff(x).subs(x, 1): 0}) == [
Eq(f(x), -sqrt(x - S.Half) / x),
Eq(f(x),
sqrt(x - S.Half) / x)
]
eq = cos(f(x)) - (x * sin(f(x)) - f(x)**2) * f(x).diff(x)
assert dsolve(eq, f(x), ics={f(0): 1}, hint='1st_exact',
simplify=False) == Eq(x * cos(f(x)) + f(x)**3 / 3,
Rational(1, 3))
assert dsolve(eq, f(x), ics={f(0): 1}, hint='1st_exact',
simplify=True) == Eq(x * cos(f(x)) + f(x)**3 / 3,
Rational(1, 3))
assert solve_ics([Eq(f(x), C1 * exp(x))], [f(x)], [C1], {f(0): 1}) == {
C1: 1
}
assert solve_ics([Eq(f(x),
C1 * sin(x) + C2 * cos(x))], [f(x)], [C1, C2], {
f(0): 1,
f(pi / 2): 1
}) == {
C1: 1,
C2: 1
}
assert solve_ics([Eq(f(x),
C1 * sin(x) + C2 * cos(x))], [f(x)], [C1, C2], {
f(0): 1,
f(x).diff(x).subs(x, 0): 1
}) == {
C1: 1,
C2: 1
}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1}) == \
{C2: 1}
# Some more complicated tests Refer to PR #16098
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x, 1):0})) == \
{Eq(f(x), 0), Eq(f(x), x ** 3 / 6 - x / 2)}
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0})) == \
{Eq(f(x), 0), Eq(f(x), C2*x + x**3/6)}
K, r, f0 = symbols('K r f0')
sol = Eq(
f(x),
K * f0 * exp(r * x) / ((-K + f0) * (f0 * exp(r * x) / (-K + f0) - 1)))
assert (dsolve(Eq(f(x).diff(x),
r * f(x) * (1 - f(x) / K)),
f(x),
ics={f(0): f0})) == sol
#Order dependent issues Refer to PR #16098
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(x).diff(x).subs(x,0):0, f(0):0})) == \
{Eq(f(x), 0), Eq(f(x), x ** 3 / 6)}
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x,0):0})) == \
{Eq(f(x), 0), Eq(f(x), x ** 3 / 6)}
# XXX: Ought to be ValueError
raises(
ValueError,
lambda: solve_ics([Eq(f(x),
C1 * sin(x) + C2 * cos(x))], [f(x)], [C1, C2], {
f(0): 1,
f(pi): 1
}))
# Degenerate case. f'(0) is identically 0.
raises(
ValueError, lambda: solve_ics([Eq(f(x), sqrt(C1 - x**2))], [f(x)],
[C1], {f(x).diff(x).subs(x, 0): 0}))
EI, q, L = symbols('EI q L')
# eq = Eq(EI*diff(f(x), x, 4), q)
sols = [
Eq(f(x), C1 + C2 * x + C3 * x**2 + C4 * x**3 + q * x**4 / (24 * EI))
]
funcs = [f(x)]
constants = [C1, C2, C3, C4]
# Test both cases, Derivative (the default from f(x).diff(x).subs(x, L)),
# and Subs
ics1 = {
f(0): 0,
f(x).diff(x).subs(x, 0): 0,
f(L).diff(L, 2): 0,
f(L).diff(L, 3): 0
}
ics2 = {
f(0): 0,
f(x).diff(x).subs(x, 0): 0,
Subs(f(x).diff(x, 2), x, L): 0,
Subs(f(x).diff(x, 3), x, L): 0
}
solved_constants1 = solve_ics(sols, funcs, constants, ics1)
solved_constants2 = solve_ics(sols, funcs, constants, ics2)
assert solved_constants1 == solved_constants2 == {
C1: 0,
C2: 0,
C3: L**2 * q / (4 * EI),
C4: -L * q / (6 * EI)
}
def test_issue_16146():
raises(ValueError, lambda: dsolve([f(x).diff(
x), g(x).diff(x)], [f(x), g(x), h(x)]))
raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x)]))
