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. 

Hint
Simple way
$$y''-2\dfrac {y}{x^2} = 0$$
$$x^2y''+2xy'-2xy'-2y= 0$$
$$(x^2y')'-2(xy)' = 0$$
Integrate 
$$(x^2y')-2(xy) = K_1$$
Divide by $x^4$
$$\dfrac {x^2y'-2xy}{x^4} = \dfrac {K_1}{x^4}$$
$$(\dfrac {y}{x^2})' = \dfrac {K_1}{x^4}$$
integrate again
$$\dfrac {y}{x^2} = \int \dfrac {K_1}{x^4}dx +K_2$$
$$\boxed{y(x)=\frac {K_1} x +K_2 x^2}$$
