204 lines
12 KiB
Python
204 lines
12 KiB
Python
|
from sympy.core.function import (Derivative, Function, diff)
|
||
|
|
from sympy.core.numbers import (I, Rational, pi)
|
||
|
|
from sympy.core.relational import Eq
|
||
|
|
from sympy.core.symbol import (Symbol, symbols)
|
||
|
|
from sympy.functions.elementary.exponential import (exp, log)
|
||
|
|
from sympy.functions.elementary.miscellaneous import sqrt
|
||
|
|
from sympy.functions.elementary.trigonometric import (cos, sin)
|
||
|
|
from sympy.functions.special.error_functions import (Ei, erf, erfi)
|
||
|
|
from sympy.integrals.integrals import Integral
|
||
|
|
|
||
|
|
from sympy.solvers.ode.subscheck import checkodesol, checksysodesol
|
||
|
|
|
||
|
|
from sympy.functions import besselj, bessely
|
||
|
|
|
||
|
|
from sympy.testing.pytest import raises, slow
|
||
|
|
|
||
|
|
|
||
|
|
C0, C1, C2, C3, C4 = symbols('C0:5')
|
||
|
|
u, x, y, z = symbols('u,x:z', real=True)
|
||
|
|
f = Function('f')
|
||
|
|
g = Function('g')
|
||
|
|
h = Function('h')
|
||
|
|
|
||
|
|
|
||
|
|
@slow
|
||
|
|
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), {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) - 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_checksysodesol():
|
||
|
|
x, y, z = symbols('x, y, z', cls=Function)
|
||
|
|
t = Symbol('t')
|
||
|
|
eq = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t)))
|
||
|
|
sol = [Eq(x(t), 9*C1*exp(-6*sqrt(3)*t) + 9*C2*exp(6*sqrt(3)*t)), \
|
||
|
|
Eq(y(t), -6*sqrt(3)*C1*exp(-6*sqrt(3)*t) + 6*sqrt(3)*C2*exp(6*sqrt(3)*t))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t)))
|
||
|
|
sol = [Eq(x(t), 4*C1*exp(t*(-sqrt(1713)/2 + Rational(43, 2))) + 4*C2*exp(t*(sqrt(1713)/2 + \
|
||
|
|
Rational(43, 2)))), Eq(y(t), C1*(-sqrt(1713)/2 + Rational(39, 2))*exp(t*(-sqrt(1713)/2 + \
|
||
|
|
Rational(43, 2))) + C2*(Rational(39, 2) + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + Rational(43, 2))))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t)))
|
||
|
|
sol = [Eq(x(t), (C1*sin(sqrt(7)*t/2) + C2*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2))), \
|
||
|
|
Eq(y(t), ((C1/2 - sqrt(7)*C2/2)*sin(sqrt(7)*t/2) + (sqrt(7)*C1/2 + \
|
||
|
|
C2/2)*cos(sqrt(7)*t/2))*exp(t*Rational(3, 2)))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
|
||
|
|
sol = [Eq(x(t), C1*exp(t*(-sqrt(6) + 3)) + C2*exp(t*(sqrt(6) + 3)) - \
|
||
|
|
Rational(22, 3)), Eq(y(t), C1*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) + C2*(2 + \
|
||
|
|
sqrt(6))*exp(t*(sqrt(6) + 3)) - Rational(5, 3))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
|
||
|
|
sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - Rational(58, 3)), \
|
||
|
|
Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - Rational(185, 3))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
|
||
|
|
sol = [Eq(x(t), (C1*exp(Integral(2, t).doit()) + C2*exp(-(Integral(2, t)).doit()))*\
|
||
|
|
exp((Integral(5*t, t)).doit())), Eq(y(t), (C1*exp((Integral(2, t)).doit()) - \
|
||
|
|
C2*exp(-(Integral(2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (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)))
|
||
|
|
sol = [Eq(x(t), (C1*cos((Integral(t**2, t)).doit()) + C2*sin((Integral(t**2, t)).doit()))*\
|
||
|
|
exp((Integral(5*t, t)).doit())), Eq(y(t), (-C1*sin((Integral(t**2, t)).doit()) + \
|
||
|
|
C2*cos((Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (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)))
|
||
|
|
sol = [Eq(x(t), (C1*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
|
||
|
|
C2*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit())), \
|
||
|
|
Eq(y(t), (C1*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()) + \
|
||
|
|
C2*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
|
||
|
|
root0 = -sqrt(-sqrt(47) + 7)
|
||
|
|
root1 = sqrt(-sqrt(47) + 7)
|
||
|
|
root2 = -sqrt(sqrt(47) + 7)
|
||
|
|
root3 = sqrt(sqrt(47) + 7)
|
||
|
|
sol = [Eq(x(t), 43*C1*exp(t*root0) + 43*C2*exp(t*root1) + 43*C3*exp(t*root2) + 43*C4*exp(t*root3)), \
|
||
|
|
Eq(y(t), C1*(root0**2 - 5)*exp(t*root0) + C2*(root1**2 - 5)*exp(t*root1) + \
|
||
|
|
C3*(root2**2 - 5)*exp(t*root2) + C4*(root3**2 - 5)*exp(t*root3))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
|
||
|
|
root0 = -sqrt(-sqrt(109)/2 + Rational(15, 2))
|
||
|
|
root1 = sqrt(-sqrt(109)/2 + Rational(15, 2))
|
||
|
|
root2 = -sqrt(sqrt(109)/2 + Rational(15, 2))
|
||
|
|
root3 = sqrt(sqrt(109)/2 + Rational(15, 2))
|
||
|
|
sol = [Eq(x(t), 3*C1*exp(t*root0) + 3*C2*exp(t*root1) + 3*C3*exp(t*root2) + 3*C4*exp(t*root3) - Rational(181, 29)), \
|
||
|
|
Eq(y(t), C1*(root0**2 - 8)*exp(t*root0) + C2*(root1**2 - 8)*exp(t*root1) + \
|
||
|
|
C3*(root2**2 - 8)*exp(t*root2) + C4*(root3**2 - 8)*exp(t*root3) + Rational(183, 29))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
|
||
|
|
sol = [Eq(x(t), C1*cos(t*(Rational(9, 2) + sqrt(109)/2)) + C2*sin(t*(Rational(9, 2) + sqrt(109)/2)) + \
|
||
|
|
C3*cos(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*sin(t*(-sqrt(109)/2 + Rational(9, 2)))), Eq(y(t), -C1*sin(t*(Rational(9, 2) + sqrt(109)/2)) \
|
||
|
|
+ C2*cos(t*(Rational(9, 2) + sqrt(109)/2)) - C3*sin(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*cos(t*(-sqrt(109)/2 + Rational(9, 2))))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
|
||
|
|
I1 = sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erfi(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(3*sqrt(7)*t**2/2)/t
|
||
|
|
I2 = -sqrt(6)*7**Rational(1, 4)*sqrt(pi)*erf(sqrt(6)*7**Rational(1, 4)*t/2)/2 - exp(-3*sqrt(7)*t**2/2)/t
|
||
|
|
sol = [Eq(x(t), C3*t + t*(9*C1*I1 + 9*C2*I2)), Eq(y(t), C4*t + t*(3*sqrt(7)*C1*I1 - 3*sqrt(7)*C2*I2))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (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)))
|
||
|
|
sol = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
|
||
|
|
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t)))
|
||
|
|
sol = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
|
||
|
|
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
|
||
|
|
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t))))
|
||
|
|
sol = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
|
||
|
|
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
|
||
|
|
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t)))
|
||
|
|
sol = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
|
||
|
|
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
|
||
|
|
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t)))
|
||
|
|
sol = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), \
|
||
|
|
Eq(y(t), C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), Eq(z(t), C1*exp(2*t) + 5*C2*cos(t) + 5*C3*sin(t))]
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
|
||
|
|
sol = [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 checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
|
||
|
|
sol = [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 checksysodesol(eq, sol) == (True, [0, 0])
|
||
|
|
|
||
|
|
eq = (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))
|
||
|
|
sol = {Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)}
|
||
|
|
assert checksysodesol(eq, sol) == (True, [0, 0])
|