Originally Posted by
HallsofIvy 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 "$\displaystyle Acos(\omega t)+ B sin(\omega t)$" rather tham $\displaystyle Ae^{i\omega t}$ because your equation uses "sin(t)" rather than a complex exponential.
(Of course, $\displaystyle e^{i\omega t}= cos(\omega t)+ sin(\omega t)$.)
For part iii, that won't work because, with $\displaystyle \omega^2= 400$, "$\displaystyle sin(\omega t)$" and "$\displaystyle cos(\omega t)$" are already solutions to the associated homogeneous equation. Do you remember what to do in that case?