How do you derive a power series from an initial value problem $y''(x)=xy(x)$

Question

I have an initial value problem of $y''(x)=xy(x)$ , $y(0)=1$, $y'(0)= 1$

I don't know how to derive the power series description of the initial value.

Could someone show me how I would go about doing this?

Answer 1

Write $y=\sum_{n\ge 0}c_n x^n$. The initial conditions give $c_0=c_1=1$. The ODE gives a recursion relation on the $c_n$.