Finding the Homogenous Solution to a Variable Coefficients 2nd order ODE

[joelio36]

x y'' + (x + 1) y' = 2 x

Solve for y(x).

Due to the coefficients being a function of x, I have no idea where to start to find the homogenous solution (Complementary Function). I know how to proceed after this part with the variation of parameters method.

I just have no idea where to begin to find a solution of this equation, which seems to be the pre-requisite for all solution methods! Do you just used ansatz? If so how would I pick an ansatz?

I'm tearing my hair out here, any help would be greatly appreciated.

 

[fzero]

The homogenous equation x y'' + (x + 1) y' = 0 can be solved by separation of variables.

 

[gomunkul51]

y' = z
then separation of variables in a first order ODE.