def solve_aux_eq(numa, dena, numy, deny, x, m):
"""
Helper function to find a polynomial solution
of degree m for the auxiliary differential
equation.
"""
# Assume that the solution is of the type
# p(x) = C_0 + C_1*x + ... + C_{m-1}*x**(m-1) + x**m
psyms = symbols(f'C0:{m}', cls=Dummy)
K = ZZ[psyms]
psol = Poly(K.gens, x, domain=K) + Poly(x**m, x, domain=K)
# Eq (5.16) in Thesis - Pg 81
auxeq = (dena*(numy.diff(x)*deny - numy*deny.diff(x) + numy**2) - numa*deny**2)*psol
if m >= 1:
px = psol.diff(x)
auxeq += px*(2*numy*deny*dena)
if m >= 2:
auxeq += px.diff(x)*(deny**2*dena)
if m != 0:
# m is a non-zero integer. Find the constant terms using undetermined coefficients
return psol, linsolve_dict(auxeq.all_coeffs(), psyms), True
else:
# m == 0 . Check if 1 (x**0) is a solution to the auxiliary equation
return S.One, auxeq, auxeq == 0
