Hint: treat each time interval separately, and match the solutions in the different intervals by requiring u and u' to be continuous functions of t.
I started this problem by trying to solve for the complimentary solution. The characteristic equation is r^2 + 1 =0. Meaning r = i.
Thus, I got:
Using the initial conditions, I got c1 and c2 =0. I don't think this is right though...