Solution of Differential equation by series

Question

Excuse me , can you help me on solving the following differential equation $y''(x) - 1/(x+A)y'(x)+C/(x+B) y=0$

I use the series but i stop on a point and then can not complete specially on finding the recurrence relation

Answer 1

Assuming that the equation is $$y'' -\frac {y'}{x+a}+\frac {c\,y}{x+b} =0$$ there is no explicit solution except if $a=b=0$. If this was the case, one solution would be $$y=Kc x J_2\left(2 \sqrt{c} \sqrt{x}\right)$$ where appears the Bessel function of the first kind which has a series representation.

So, let $$y=\sum_{i=0}^n d(i)\, x^i \quad y'=\sum_{i=0}^n i\,d(i)\, x^{i-1} \quad y''=\sum_{i=0}^n i(i-1)\,d(i)\, x^{i-2} $$ Replace, use the long division and group terms. Now, cancel the coefficients.

It will effectively be quite tedious but it is doable. For example, we need to solve (one at the time would be the best) $$\frac{d(1)}{a^2}-\frac{2 d(2)}{a}-\frac{c d(0)}{b^2}+\frac{c d(1)}{b}+6 d(3)=0$$ $$-\frac{d(1)}{a^3}+\frac{2 d(2)}{a^2}-\frac{3 d(3)}{a}+\frac{c d(0)}{b^3}-\frac{c d(1)}{b^2}+\frac{c d(2)}{b}+12 d(4)=0$$ and so on.