General solution of $y''-2xy'+2y=0$

Question

I have to find the general solution of the differential equation $$y''-2xy'+2y=0.$$ An obvious solution is $y(x)=ax$ but I am unable to find another solution. Any hint as to how to proceed or which kind of method to apply is greatly appreciated.

Answer 1

Hints: there is a standard method called variation of parameters. Take $y=xg(x)$ (obtained by replacing the constant in your first solution by a function). Plug this into the ODE and put $h=g'$. You will get a first order ODE for $h$. You should now be able to complete the answer. Note: please see comment below by LutzL.

Answer 2

Making $z = x y$ we obtain

$$ z''-\frac{2(x^2-1)}{x}z' = 0 $$

now making $v = z'$

$$ v'-\frac{2(x^2-1)}{x}v = 0 $$

which is separable.