Answer to Question #295146 in Differential Equations for Bala

Question

Find a power series solution pf xy'=y

Answer 1

CORRECTED SOLUTION.

Let $y=\sum\limits_{n=0}^{+\infty}c_nx^n$. Then $xy'=x\sum\limits_{n=0}^{+\infty}nc_nx^{n-1}=\sum\limits_{n=0}^{+\infty}nc_nx^n$.

Since $xy'=y$, it must be $nc_n=c_n$ for all $n=0,1,2,\dots$

This means $c_n=0$ for all $n\ne 1$ and $c_1$ can be arbitrary.

Answer. $y=c_1x$.