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## Homework Statement

The question I am working on is the one in the file attached.

## Homework Equations

y = u

_{1}y

_{1}+ u

_{2}y

_{2}:

u

_{1}'y

_{1}+ u

_{2}'y

_{2}= 0

u

_{1}'y

_{1}' + u

_{2}'y

_{2}' = g(t)

## The Attempt at a Solution

I think I have got part (i) completed, with y

_{1}= e

^{3it}and y

_{2}= e

^{-3it}. This gives a general solution to the homogeneous equation to be y = C

_{1}cos3t + C

_{2}sin3t.

For part (ii) I know that for variation of parameters you need to substitute y

_{1}and y

_{2}and their derivatives into the above system of equations to solve for u

_{1}' and u

_{2}', then integrate these to find u

_{1}and u

_{2}, from which you can get the desired solution of

y = y

_{p}+ y

_{h}

But I find that when I try to do that by substituting y

_{1}= e

^{3it}and y

_{2}= e

^{-3it}I just end up with very complicated expressions involving lots of e's and i's which the desired solution does not contain. I know it involves combining the information about the general soln from (i) with the method I have described, but I just don't know how to make it work. Any help would be appreciated, thank you in advance!