Find series expansion of the solutions to the following DE about $x = 0$. Try to sum in closed form any infinite series that appear:
$ y'' + (e^x - 1)y = 0 $
My approach: OF course $x = 0$ is ordinary point, so we can find the taylor expansion of the solution assuming $y = \sum_{n=0}^{\infty} c_nx^n $ is a solution. And so we differentiale twice this expression and put it back into the DE to obtain a nasty equation like this: (after shifting indices)
$ \sum_{n=0}^{\infty} c_{n+2}(n+1)(n+2)x^n + \sum_{n=0}^{\infty} \frac{1}{n!}x^n \sum_{n=0}^{\infty} c_nx^n - \sum_{n=0}^{\infty} c_nx^n = 0 $ My question is: is this procedure fine? or is there a better way to approach this problem? and how can I find the $c_i$ in a economical manner?