The question I am stuck on is from Strauss (Exercise 4.3.10):
Solve the wave equation with Robin boundary conditions under the assumption that $a_0+a_l<-a_0a_ll$ holds.
I know that this condition means there is one negative eigenvalue $\lambda_0$ and the rest are positive. I should be able to compute $\lambda_0$ from $$\tanh(\sqrt{\lambda_0}l)=\frac{-(a_0+a_l)\sqrt{\lambda_0}}{\lambda_0+a_0a_l}$$ and the corresponding eigenfunction from $$X_0(x)=\cosh(\sqrt{\lambda_0}x)+\frac{a_0}{\sqrt{\lambda_0}} \sinh(\sqrt{\lambda_0}x)$$ but I don't know how to do this. Once I find $X_0(x)$ I believe I can substitute it in to $$u(x,t)=\sum_{n=0}^\infty X_n(x) T_n(t)$$ along with $T_0(t)=A_0 \cosh(\sqrt{|\lambda_0|}ct)+B_0 \sinh(\sqrt{|\lambda_0|}ct)$ to get a solution. But I have no clue if this is correct. Any help would be greatly appreciated.