# Frobenius method for linear second order differential equation

I am trying to solve $x^{2}y''+xy'-9y=0$ using Frobenius' method. Plugging in the derivatives of $y=x^{s}\sum ^{\infty }_{i=0}c_{i}x^{i}$ to the equation, I get, for the lowest power $x^s$, the indicial equation $s(s-1)c_{0}+sc_{0}-9c_{0}$ which yields $s=\pm3$.

I can also solve the equation given by the coefficients of $x^{s+1}$ and $x^{s+2}$ as giving $c_{1}=0$ and $c_{2}=0$.

But when I do the general term $x^{s+n}$ I get $(nc_{n})(2s+1)+n(n-1)c_{n}=0$, which does not specify $c_{n}$.

Without a recurrence relation, I do not know what is happening, nor how to proceed with the solution to the equation.

Why does a recurrence relation not manifest from the general term?

If what you calculated is ok for $s$ which I call $r$, then a solution for $r=3$ is the following. Do the same for $r=-3$.