Differential Equation Help Please MFP3 AQA

I think you may be a little confused about the difference between particular integrals and complementary functions. If $\lambda _1$ and $\lambda _2$ are solutions of the auxiliary equation, then the CF of the ODE is $Ae^{\lambda _1x} + Be^{\lambda _2x}$. In basic terms, the complementary function is supposed to be such that the LHS is zero when it is plugged in to the ODE. Therefore, I have no idea how you decided that y = Csin2x + Dcos2x was the CF.

The particular integral, on the other hand, is such that if you plug it into the LHS, you should get out the RHS. The question gives you the form of trial function (namely $y=ax\cos 2x$) so all you have to do is plug it in and solve for the constant a. I think the reason that the question asks you just to find the constant a which fits their PI is because it doesn't work for the usual trial function (for one reason or another that I'm not quite sure about).

EDIT: I don't think I've helped at all though, you'd be better off waiting for someone else's advice as my brain isn't in gear today.