Translate the following into a PDE with boundary conditions.

A horizontal string has lenth $L$. When $t<0$ the string is held in place so the point $x$ units from its left end has vertical displacement $\sin\!\left(\frac{\pi x}{L}\right)$. The left end of the string is secured to the x-axis, but starting at time $t=0$ the part between its ends is released and its right end is forced to move with vertical displacement $\sin\!\left(\pi t\right)$. At each point its vertical motion is damped with a force proportional to its instantaneous vertical velocity. Use c as the constant of proportionality, and assume the mass density (mass per unit length) of the string is 1. How to enter the answers.

The PDE is
When $t\geq 0$ the boundary values / initial conditions are:
Left end is secured:
Right end moves vertically:
Initially at rest:
Initial displacement is $\sin\!\left(\frac{\pi x}{L}\right)$: