Second order non homogeneous ODE, IVP

[Cocoleia]

Homework Statement

I need to solve:
x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9

Homework Equations

The Attempt at a Solution

I know that the answer is: y=x^2+2x^3+x^3lnx
Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...

 

[LCKurtz]

I haven't checked your work because your image is difficult to read. Yes, variation of parameters can be used for a particular solution. You can also get a particular solution by undetermined coefficients similar to how you would do for a constant coefficient DE. The following link is to a pdf file that explains how to construct a particular solution $y_p$.
http://www.maa.org/sites/default/files/pdf/cmj_ftp/CMJ/May 2010/4 Capsules/1 DeLeon/ecexp_clscap2.pdf