What's the power series solution of $y' = x^2\,y$

Question

My attempted solution

I'm not sure how to factor out $x$. In the other problems I've solved, both series terms have the same power of $x$. but in this problem one series contains $x^{n+2}$ and the other contains $x^n$.

Answer 1

Make a change of variable in the summation index. For instance, letting $n+2=k$ we have $$ \sum_{n=0}^\infty c_nx^{(n+2)}=\sum_{k=2}^\infty c_{k-2}x^k. $$