Question

For what values of n is y=x^n a solution of the differential equation \[x^2y\prime\prime+xy\prime-y=0\]

Answer 1

\[
x^2 y''(x)+x y'(x)-y(x)=0\\
(n-1) n x^n+n x^n-x^n=0\\
(n-1) (n+1) x^n=0\\
n=1\\
n=-1
\]