Linear homogenous second order ODE without constant coefficients I am having trouble finding the general solution of the following second order ODE for $y = y(x)$ without constant coefficients: 
$3x^2y'' = 6y$
$x>0$
I realise that it may be possible to simply guess the form of the solution and substitute it back into the the equation but i do not wish to use that approach here. 
I would appreciate any help, thanks. 

From the form of this differential equation $$3x^2y'' = 6y$$ it is evident that $y=x^k$ is a potential solution.
Upon substitution of $y=x^k$ we come up with $k=2$ and $k=-1$
Thus the general solution is $$ y= C_1 x^2 +C_2 x^{-1}.$$
