Quite stumped with this one so far. I have the following non-homogeneous ODE:
And I need to find a solution for $x_0<0$ using Frobenius. Obviously we can center the solution about $x_0=0$, and we can see it is a regular singular point based on the coefficients of $y'$ and $y$. But I searched quite a bit and couldn't find anything about the non-homogeneous part of the equation (every example that I find on here either is homogeneous, or the NH part is a constant). I know for $p(x)$ and $q(x)$ we check for RSP such that:
Which are analytical for $x_0=0$, so no problem there. But I have no clue what to do with the non-homogeneous part of the equation to verify that it is really a RSP. Of course the problem would easily be solved simply with a regular power series, but I absolutely have to use Frobenius without using series to check for analyticity. Any pointers?
Apologies if the terminology is a bit off, english is not my mother tongue. Thanks in advance!