I'm working on my final project in this independent study I'm in. I worked out this problem, but I want to verify that I'm right.

A string of length L with both endpoints fixed, such as the string of a piano, is struck at its midportion, the impact thus imparting to the string an initial velocity which is defined by:

Find the motion of the string.

I set up the following boundary value problem:

and I got the following Fourier Series Solution:

Does this look right? I would appreciate any help!!!