1D wave equation (PDE)

[hippolyta]

Hi everyone,

I'm a 3rd year Engineering Science student and at the moment we're learning how to solve Partial Differential Equations, in particular the 1D wave equation.

Here's the thing though:

The BC's are the usual, u(0,t)=0 and u(L,t)=0.
The IC's are u'(x,0)=0 and, importantly, u(x,0)=(some long sin/cos expression)

I am at the stage where I've found the general equation for u(x,t), with two constants, F and G.
But when I use orthogonality to get the equation into the form it should be in in order to find F, I am forced to use integration by parts on two rather complex sin/cos expressions, together. Because neither of these reduces (or will ever reduce) to a constant term, it seems I can never get to the end of this integral! Am I doing something fundamentally wrong? It seems so..either I'm overlooking a way of solving the integral, or I shouldn't be using integration by parts in the first place...I've read over the notes several times, but the problem is that in all the examples we're given, the IC is a linear expression in terms of x, so that it becomes a constant when it's differentiated, and hence integration by parts is possible.
Any suggestions would be greatly appreciated.

Thanks

[hippolyta]

Oh and I forgot to mention that the two rather elaborate sin/cos expressions are not such that I can differentiate/integrate one to get the other, so I can't use that approach - this is probably important