Solving inhomogenous wave equation

[wahoo2000]

Aha.. but then
u(1,0)= -\frac{32}{\pi^{4}} \sum_{m=0}^{\infty} \frac{(-1)^{m}}{(2m+1)^{4}}*-(-1)^{m}*1-\frac{1}{6}+\frac{1}{2}+1=1<br /> \Rightarrow \sum_{m=0}^{\infty} \frac{1}{(2m+1)^{4}}=(1-\frac{4}{3})*(-\frac{\pi^{2}}{32})=\frac{\pi^{4}}{96}<br /> <br />

 

[gabbagabbahey]

Yep, and

\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{4}} =\frac{\pi^{4}}{96}

can be used to answer your question by substituting n=m+1 :

\Rightarrow \sum_{n=1}^{\infty} \frac{1}{(2n-1)^{4}} = \sum_{m+1=1}^{\infty} \frac{1}{(2(m+1)-1)^{4}}<br /> = \sum_{m=0}^{\infty} \frac{1}{(2m+1)^{4}}=\frac{\pi^{4}}{96}

 

[wahoo2000]

Thank you SO much for your patience. Tomorrow I am going through this problem from the start again to make sure I get each step.

 

[gabbagabbahey]

Thank you SO much for your patience. Tomorrow I am going through this problem from the start again to make sure I get each step.

Your welcome. The critical point of the problem was the part about the boundary conditions for w(x,t). Choosing the right boundary conditions (by choosing boundary conditions for S) makes the solution much easier. In general, you should try to set the conditions on S such that w(x,t) vanishes at one endpoint (x=0 in this case).