He is talking about the "method of undetermined coefficients". If you are expected to be able to solve a problem like this, you must have been taught that. It might be the complex exponential that is confusing you. I would recommend something like "
" rather tham
because your equation uses "sin(t)" rather than a complex exponential.
(Of course,
.)
For part iii, that won't work because, with
, "
" and "
" are already solutions to the associated homogeneous equation. Do you remember what to do in that case?