EDIT: There was a silly typo, $y^n$ is actually $y''$... sorry everyone!
In that case just finding $y''$ and substituting this and $y$ into the left side will yield the answer 0.
I have noted that $y^n$ refers to the nth derivative.
I found that the 1st, 5th, 9th, ..., $(4m+1)$th derivatives have a sign pattern of - and +, respectively, in front of each trig ratio.
That is, $y^n = -Ak^n\sin kt + Bk^n\cos kt$.
The 2nd, 6th, 10th, ..., $(4m+2)$th derivates have a sign pattern of - and -.
The 3rd, 7th, 11th, ... $(4m+3)$th derivates have a sign pattern of + and -.
Finally, the 4th, 8th, 12th, ..., $(4m+4)$th derivatives have a sign pattern of + and +.
I am not sure if this is relevant in breaking down the proof into cases...
Thank you in advance for any insight.