def test_solve_poly_rde_no_cancel():
# deg(b) large
DE = DifferentialExtension(extension={"D": [Poly(1, x), Poly(1 + t ** 2, t)]})
assert solve_poly_rde(Poly(t ** 2 + 1, t), Poly(t ** 3 + (x + 1) * t ** 2 + t + x + 2, t), oo, DE) == Poly(t + x, t)
# deg(b) small
DE = DifferentialExtension(extension={"D": [Poly(1, x)]})
assert solve_poly_rde(Poly(0, x), Poly(x / 2 - S(1) / 4, x), oo, DE) == Poly(x ** 2 / 4 - x / 4, x)
DE = DifferentialExtension(extension={"D": [Poly(1, x), Poly(t ** 2 + 1, t)]})
assert solve_poly_rde(Poly(2, t), Poly(t ** 2 + 2 * t + 3, t), 1, DE) == Poly(t + 1, t, x)
# deg(b) == deg(D) - 1
DE = DifferentialExtension(extension={"D": [Poly(1, x), Poly(t ** 2 + 1, t)]})
assert no_cancel_equal(Poly(1 - t, t), Poly(t ** 3 + t ** 2 - 2 * x * t - 2 * x, t), oo, DE) == (
Poly(t ** 2, t),
1,
Poly((-2 - 2 * x) * t - 2 * x, t),
)
def test_solve_poly_rde_no_cancel():
# deg(b) large
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
oo, DE) == Poly(t + x, t)
# deg(b) small
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert solve_poly_rde(Poly(0, x), Poly(x/2 - S(1)/4, x), oo, DE) == \
Poly(x**2/4 - x/4, x)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
Poly(t + 1, t, x)
# deg(b) == deg(D) - 1
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert no_cancel_equal(Poly(1 - t, t),
Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
(Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))