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# Fun diff eq problem watch

1. Strictly speaking, I am not needing help on this. I solved it as part of an assignment that was turned in yesterday. Just thought I'd post it because it was rather involved (unless there's an easier method than the one I used) and strangely fun.

Yay for Laplace transforms and step functions.
2. I'm useless at differential equations, but I think...

Spoiler:
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...you should be able to solve the homogeneous equation to get linear indep. solutions e^pt and e^qt, and then find a particular solution, which will be of the form Rsin(t + a) between 0 and pi and will be constant afterwards. Then for 0 <= t < pi, we have y = Ae^pt + Be^qt + Rsin(t+a), and for t >= pi we have y = Ce^pt + De^qt + E. Use the initial conditions to solve for A and B, then use continuity of y, y' and y'' at t = pi to solve for C, D, E. (It'll probably come out as C = A, D = B, E = Rsin(pi+a), but I'm not sure.)
3. Dealing with g(t) all at once would be a cleaner method, I think.

Transform both sides while employing the "nth derivative" property found here on the left side and then lots and lots of algebra.
4. And what if, like me, you don't know Laplace transforms, and want to solve it by A-level methods?

6. I always knew you were odd.
7. Being odd is more fun. Cosine? Come on...

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Updated: August 5, 2009
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