In part (a) we found a solution for a specific f(r) that happened to be one of the original eigenfunctions and so we were able to compare just by analyzing the coefficient of one term. However now we have a general f(r) and if we continue from part (a) we have,
- .
We wish to impose an orthogonality argument so that we can isolate each individual coefficient, . Recall that if we have a problem in Sturm-Liouville form,
then the eigenfunctions have the orthogonality relationship
Our eigenfunction problem for r was
which we can write in Sturm-Liouville form to get
so our orthogonality relation is
If we compute the integrals we can use a substitution of
to get
We immediately recognize this new integral as the standard sine orthogonality relation and so we get
Returning to
we can use our orthogonality relation to get
Therefore we now have the constants for any and therefore we have the solution to the partial differential equation,