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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!
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