The Attempt at a Solution
First issue is I was under the impression that a particular solution is the final solution to a DE; a solved DE with initial conditions applied, but it would be weird for that to be the first part of the question so I'm interpreting a) as asking for the particular integral (maybe they're the same thing :S). Anyway for the first part of the question I've done:
so dyPI/dt = iCeit and d2yPI/dt2= -Ceit
which gives: C=1/3
Using a trial solution for the complementary function gives yCF=Ae2it+Be-2it
and a general solution of:
At this point I have no idea how to start applying initial conditions as I have an expression that's full of 'i's, so maybe I don't understand the first part of the question and have gone about the whole thing the wrong way. Some help would be really appreciated!