solve differential equation

by using the substitution x=e^z ,
solve the euler differnetial equation

x^2 d^2y/dx^2 + x dy/dx + y = 0
Solution
Please also see http://mathworld.wolfram.com/EulerDifferentialEquation.html
for reference
x^2 d^2y/dx^2 + x dy/dx + y = 0
d^2y/dx^2 + (1/x) dy/dx + y/x^2 = 0
Here α=1 β=1
let x=e^z
The equation will becomes
d^2y/dz^2 + A dy/dx + By = 0
where A=α-1=0, B=β=1
So the equation is
d^2y/dz^2 + y = 0
The characteristic equation is
λ^2+1=0
λ=i or -i
So the answer is
y=Acos z+Bsin z=Acos (ln x)+Bsin (ln x) where A,B are constant