Variation of Parameter Problem

[bengaltiger14]

Homework Statement

Find the general solution using the method of variation of parameters of:

y''-6y'+9y=(x^-3)(e^3x)

I found the roots of the corresponding homogeneous equation to be lamba = 3. So there are repeated roots. My question is, how do I solve a variation of parameter question with repeated roots? I know how to do it using reduction of order but confused on variation of parameters

 

[HallsofIvy]

The same way you solve it without repeated roots! Since x= -3 is a double root of the characteristic equation, two independent solutions to the associated homogeneous equation are $e^{-3x}$ and $xe^{-3x}$.

Look for a solution to the entire equation of the form $y(x)= u(x)e^{-3x}+ v(x)xe^{-3x}$. Now just follow the usual procedure for variation of parameters.