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We use the substitution:
Then, the equation becomes:
The initial conditions become:
And we get periodic boundary conditions:
We start with the process of separating the equation:
Then, substituting it back in the homogenous wave equation we get:
Dividing this by XT completes the separation:
Since both sides are totally independent, the only way for the equation above to be true is if both sides equal the same constant:
Since we have periodic boundary conditions, the only non trivial solution for the spatial equation is occurs when and the solution is:
Now for the temporal equation:
Now we assume a particular solution which is an expansion of the spatial eigenfunctions with time dependent coefficients:
Also, we shall expand the non-homogenous term with the eigenfunctions:
Integrating by parts:
So we ...
Neumann and Dirichlet Problems are solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.