def test_constant_multiple():
assert constant_renumber(constantsimp(C1 * 2 + 2, [C1])) == C1
assert constant_renumber(constantsimp(x * 2 / C1, [C1])) == C1 * x
assert constant_renumber(constantsimp(C1**2 * 2 + 2, [C1])) == C1
assert constant_renumber(constantsimp(sin(2 * C1) + x + sqrt(2),
[C1])) == C1 + x
assert constant_renumber(constantsimp(2 * C1 + C2, [C1, C2])) == C1
def test_constant_power_as_base():
assert constant_renumber(constantsimp(C1**C1, [C1])) == C1
assert constant_renumber(constantsimp(Pow(C1, C1), [C1])) == C1
assert constant_renumber(constantsimp(C1**C1, [C1])) == C1
assert constant_renumber(constantsimp(C1**C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2**C1, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2**C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1**y, [C1, y])) == C1
assert constant_renumber(constantsimp(C1**x, [C1])) == C1**x
assert constant_renumber(constantsimp(C1**2, [C1])) == C1
assert constant_renumber(constantsimp(C1**(x * y), [C1])) == C1**(x * y)
def test_constant_function():
assert constant_renumber(constantsimp(sin(C1), [C1])) == C1
assert constant_renumber(constantsimp(f(C1), [C1])) == C1
assert constant_renumber(constantsimp(f(C1, C1), [C1])) == C1
assert constant_renumber(constantsimp(f(C1, C2), [C1, C2])) == C1
assert constant_renumber(constantsimp(f(C2, C1), [C1, C2])) == C1
assert constant_renumber(constantsimp(f(C2, C2), [C1, C2])) == C1
assert constant_renumber(constantsimp(f(C1, x), [C1])) == f(C1, x)
assert constant_renumber(constantsimp(f(C1, y), [C1, y])) == C1
assert constant_renumber(constantsimp(f(y, C1), [C1, y])) == C1
assert constant_renumber(constantsimp(f(C1, y, C2), [C1, C2, y])) == C1
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 test_constant_add():
assert constant_renumber(constantsimp(C1 + C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 + 2, [C1])) == C1
assert constant_renumber(constantsimp(2 + C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 + y, [C1, y])) == C1
assert constant_renumber(constantsimp(C1 + x, [C1])) == C1 + x
assert constant_renumber(constantsimp(C1 + C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 + C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2 + C1, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1 + C2 + C1, [C1, C2])) == C1
def test_ode_solutions():
# only a few examples here, the rest will be tested in the actual dsolve tests
assert constant_renumber(constantsimp(C1*exp(2*x) + exp(x)*(C2 + C3), [C1, C2, C3])) == \
constant_renumber(C1*exp(x) + C2*exp(2*x))
assert constant_renumber(
constantsimp(Eq(f(x),
I * C1 * sinh(x / 3) + C2 * cosh(x / 3)),
[C1, C2])) == constant_renumber(
Eq(f(x),
C1 * sinh(x / 3) + C2 * cosh(x / 3)))
assert constant_renumber(constantsimp(Eq(f(x), acos((-C1)/cos(x))), [C1])) == \
Eq(f(x), acos(C1/cos(x)))
assert constant_renumber(
constantsimp(Eq(log(f(x) / C1) + 2 * exp(x / f(x)), 0),
[C1])) == Eq(log(C1 * f(x)) + 2 * exp(x / f(x)), 0)
assert constant_renumber(constantsimp(Eq(log(x*sqrt(2)*sqrt(1/x)*sqrt(f(x))
/C1) + x**2/(2*f(x)**2), 0), [C1])) == \
Eq(log(C1*sqrt(x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
assert constant_renumber(constantsimp(Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(x/C1) -
cos(f(x)/x)*exp(-f(x)/x)/2, 0), [C1])) == \
Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) - cos(f(x)/x)*
exp(-f(x)/x)/2, 0)
assert constant_renumber(constantsimp(Eq(-Integral(-1/(sqrt(1 - u2**2)*u2),
(u2, _a, x/f(x))) + log(f(x)/C1), 0), [C1])) == \
Eq(-Integral(-1/(u2*sqrt(1 - u2**2)), (u2, _a, x/f(x))) +
log(C1*f(x)), 0)
assert [constantsimp(i, [C1]) for i in [Eq(f(x), sqrt(-C1*x + x**2)), Eq(f(x), -sqrt(-C1*x + x**2))]] == \
[Eq(f(x), sqrt(x*(C1 + x))), Eq(f(x), -sqrt(x*(C1 + x)))]
def find_kovacic_sol(eq):
z = list(eq.atoms(Derivative))[0].args[0]
x = list(z.free_symbols)[0]
eq = eq.expand().collect(z)
a, b, c = match_2nd_order(eq, z, x)
# Transform the differential equation to a simpler form
# using z(x) = y(x)*exp(Integral(-b/(2*a))) and find its solution
ysol = find_kovacic_simple(x, a, b, c)
zsol = list(map(lambda sol: sol*exp(Integral(-b/(2*a), x).doit()), ysol))
zsol = _remove_redundant_solutions(Eq(eq, 0), list(map(lambda sol: Eq(z, sol), zsol)), 2, x)
C1, C2 = symbols('C1 C2')
if len(zsol) == 2:
return constantsimp(Eq(z, C1*zsol[0].rhs + C2*zsol[1].rhs), [C1, C2])
if len(zsol) == 1:
sol1 = zsol[0].rhs
sol2 = sol1*Integral(exp(Integral(-b, x).doit())/sol1**2, x)
zsol.append(Eq(z, sol2))
return constantsimp(Eq(z, C1*zsol[0].rhs + C2*zsol[1].rhs), [C1, C2])
return zsol
def test_constantsimp_take_problem():
c = exp(C1) + 2
assert len(Poly(constantsimp(exp(C1) + c + c * x, [C1])).gens) == 2
def test_constant_power_as_exp():
assert constant_renumber(constantsimp(x**C1, [C1])) == x**C1
assert constant_renumber(constantsimp(y**C1, [C1, y])) == C1
assert constant_renumber(constantsimp(x**y**C1, [C1, y])) == x**C1
assert constant_renumber(constantsimp((x**y)**C1, [C1])) == (x**y)**C1
assert constant_renumber(constantsimp(x**(y**C1), [C1, y])) == x**C1
assert constant_renumber(constantsimp(x**C1**y, [C1, y])) == x**C1
assert constant_renumber(constantsimp(x**(C1**y), [C1, y])) == x**C1
assert constant_renumber(constantsimp((x**C1)**y, [C1])) == (x**C1)**y
assert constant_renumber(constantsimp(2**C1, [C1])) == C1
assert constant_renumber(constantsimp(S(2)**C1, [C1])) == C1
assert constant_renumber(constantsimp(exp(C1), [C1])) == C1
assert constant_renumber(constantsimp(exp(C1 + x), [C1])) == C1 * exp(x)
assert constant_renumber(constantsimp(Pow(2, C1), [C1])) == C1
def test_constant_mul():
# We want C1 (Constant) below to absorb the y's, but not the x's
assert constant_renumber(constantsimp(y * C1, [C1])) == C1 * y
assert constant_renumber(constantsimp(C1 * y, [C1])) == C1 * y
assert constant_renumber(constantsimp(x * C1, [C1])) == x * C1
assert constant_renumber(constantsimp(C1 * x, [C1])) == x * C1
assert constant_renumber(constantsimp(2 * C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 * 2, [C1])) == C1
assert constant_renumber(constantsimp(y * C1 * x, [C1, y])) == C1 * x
assert constant_renumber(constantsimp(x * y * C1, [C1, y])) == x * C1
assert constant_renumber(constantsimp(y * x * C1, [C1, y])) == x * C1
assert constant_renumber(constantsimp(C1 * x * y, [C1, y])) == C1 * x
assert constant_renumber(constantsimp(x * C1 * y, [C1, y])) == x * C1
assert constant_renumber(constantsimp(C1 * y * (y + 1),
[C1])) == C1 * y * (y + 1)
assert constant_renumber(constantsimp(y * C1 * (y + 1),
[C1])) == C1 * y * (y + 1)
assert constant_renumber(constantsimp(x * (y * C1), [C1])) == x * y * C1
assert constant_renumber(constantsimp(x * (C1 * y), [C1])) == x * y * C1
assert constant_renumber(constantsimp(C1 * (x * y), [C1, y])) == C1 * x
assert constant_renumber(constantsimp((x * y) * C1, [C1, y])) == x * C1
assert constant_renumber(constantsimp((y * x) * C1, [C1, y])) == x * C1
assert constant_renumber(constantsimp(y * (y + 1) * C1, [C1, y])) == C1
assert constant_renumber(constantsimp((C1 * x) * y, [C1, y])) == C1 * x
assert constant_renumber(constantsimp(y * (x * C1), [C1, y])) == x * C1
assert constant_renumber(constantsimp((x * C1) * y, [C1, y])) == x * C1
assert constant_renumber(constantsimp(C1 * x * y * x * y * 2,
[C1, y])) == C1 * x**2
assert constant_renumber(constantsimp(C1 * x * y * z,
[C1, y, z])) == C1 * x
assert constant_renumber(constantsimp(C1 * x * y**2 * sin(z),
[C1, y, z])) == C1 * x
assert constant_renumber(constantsimp(C1 * C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 * C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2 * C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1 * C1 * C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1 * x * 2**x,
[C1])) == C1 * x * 2**x
def test_constant_Eq():
# C1 on the rhs is well-tested, but the lhs is only tested here
assert constantsimp(Eq(C1, 3 + f(x) * x), [C1]) == Eq(x * f(x), C1)
assert constantsimp(Eq(C1, 3 * f(x) * x), [C1]) == Eq(f(x) * x, C1)
def test_nonlocal_simplification():
assert constantsimp(C1 + C2 + x * C2, [C1, C2]) == C1 + C2 * x
def test_constant_function_multiple():
# The rules to not renumber in this case would be too complicated, and
# dsolve is not likely to ever encounter anything remotely like this.
assert constant_renumber(constantsimp(f(C1, C1, x), [C1])) == f(C1, C1, x)
