Question

Find two power series solutions of the given differential equation about the ordinary point x = 0. y'' - y = 0

Answer 1

$y=\displaystyle\sum_{n=0}^{\infin} a_nx^n$

$y''=\displaystyle\sum_{n=2}^{\infin} n(n-1)a_nx^{n-2}$

$\displaystyle\sum_{n=2}^{\infin} n(n-1)a_nx^{n-2}-\displaystyle\sum_{n=0}^{\infin} a_nx^n=0$

$\displaystyle\sum_{n=0}^{\infin} (n+1)(n+2)a_{n+2}x^{n}-\displaystyle\sum_{n=0}^{\infin} a_nx^n=0$

$\displaystyle\sum_{n=0}^{\infin} [(n+1)(n+2)a_{n+2}-a_n]x^{n}$

$a_{n+2}=\frac{a_n}{(n+1)(n+2)}$

$a_{2k}=\frac{a_0}{(2k)!},a_{2k+1}=\frac{a_1}{(2k+1)!}$

$y(x)=a_0\displaystyle\sum_{k=0}^{\infin}\frac{x^{2k}}{(2k)!}+a_1\displaystyle\sum_{k=0}^{\infin}\frac{x^{2k+1}}{(2k+1)!}$