Answer
See below
Work Step by Step
Given: $x^2y''+xy'+25y=0$
In this case the substitution $y(x) = x^r$ yields the indicial equation
$r(r-1)+r+25=0$
Factor and solve the equation
$r^2+25=0$
It follows that two linearly independent solutions to the given differential equation are
$y_1(x)=\cos (5\ln x)\\
y_2(x)=\sin (5\ln x)$
so that the general solution is
$y(x)=c_1\cos (5\ln x)+c_2\sin (5\ln x)$
