Solving the PDE Wave Equation - A_n & B_n Terms

[PhysicsGente]

Hello, this is a problem I've been trying to do but I'm not sure it is right. Particularly the A_n and B_n terms. Thanks

https://docs.google.com/open?id=0BwZLQ_me50B8M0sxelVrbTBhYVk

 

[dextercioby]

Putting a countable infinity of A's and B's instead of only 2 values A and B is a standard technique in Fourier analysis, which is particulary well suited for linear ODE's/PDE's. That is you rightfully assume that, because the linear/vector space of solutions is infinite dimensional, there is one (generally different) coefficient for each basis vector of the solution space, hence the infinite summation.

 

[HomogenousCow]

Which part of it do you not understand.
This is how a homogenous linear PDE is usually solved:
1.separate equation (possible for many of the usual PDEs)
2.Solve the equations to get a "basis"
3.Write an infinite sum with them
4.Use the imposed initial/boundary conditions to solve for the coefficients of the linearly independent solutions with the generalized Fourier's trick

If none of that made sense to you, then you are not ready for that book.