I was rather hoping that the OP would make an attempt and say where s/he was stuck if help was still required. Short of using my editing powers, this is now impossible.
As it is, you've left only one trivial (but important) thing for the OP to do. I hope that this one small thing remains left for the OP.
yes I did get that answer, plus or minus 5/2
part b) I'm stuck on again. "For those values of k, verify that every member of the family of functions y=Asinkt+Bcoskt is also a solution." (where A,B, and of course k, are constants). This is what I did:
then I plug in the value of k and get
I'm uncertain what to do next or if the question is completed
Note my correction in red first (you had k, when it should have been t).
Now, simplify the LHS to see that LHS = RHS which implies that the family of functions satisfies the differential equation.
To answer you're last question on LaTeX, click on the edited LaTeX image to see the code.