Please help me find a value for m so that y=x^m is a solution of the given differential equation.

(x^2)y''-y=0

Question

Please help me find a value for m so that y=x^m is a solution of the given differential equation.

(x^2)y''-y=0

loser66 · Answer

since y =x$^m$ is solution, we have 
          y'= m x$^{m-1}$
          y"= m (m-1)x $^{m-2}$
    x$^2$y"= m(m-1)(m-2)x$^m$
Yours becomes m(m-1)(m-2)x$^m$-x$^m$=0
factor x $^m$ out, solve for m you have value of such that y = x$^m$ is solution.

loser66 · Answer

*value of m such that...

anonymous · Answer

@Babyslapmafro, got it?

babyslapmafro · Answer

why does (m-2) become a coefficient?

babyslapmafro · Answer

@SithsAndGiggles 
@loser66

anonymous · Answer

Minor typo, I think:
$$\begin{align*}
y&=x^m\
y'&=mx^{m-1}\
y''&=m(m-1)x^{m-2}&\Rightarrow~~x^2y''=m(m-1)x^m
\end{align*}$$
Substituting into the equation you have
$$m(m-1)x^m-x^m=0\
x^m\left(m^2-m-1\right)=0$$