Answer
See below
Work Step by Step
Given: $x^2y''+9xy'+15y=0$
In this case the substitution $y(x) = x^r$ yields the indicial equation
$r(r-1)+9r+15=0$
Factor and solve the equation
$r^2+8r+15=0$
It becomes
$(r+3)(r+5)=0$
It follows that two linearly independent solutions to the given differential equation are
$y_1(x)=x^{-3}\\
y_2(x)=x^{-5}$
so that the general solution is
$y(x)=c_1x^{-3}+c_2x^{-5}$
