4.1. Separation of variables (the first blood)


### Dirichlet-Dirichlet

For boundary value problem \begin{aligned} &u_{tt}-c^2u_{xx}=0,\\ &u|_{x=0}=u|_{x=l}=0 \end{aligned} there are simple in the form $$u(x,t)= \cos(\frac{n\pi ct}{l})\sin(\frac{n\pi x}{l}):$$

and the general solutions of $u_{tt}-c^2u_{xx}=0$, $u|_{x=0}=u|_{x=L}=0$ in the form $$u(x,t)= \sum_{n=1}^{\infty}\bigl[A_n\cos(\frac{n\pi ct}{l})+B_n\sin (\frac{n\pi ct}{l})\bigr]\sin(\frac{n\pi x}{l})$$