Translate the following into a PDE with boundary conditions.

A horizontal string of length $L$ is held stretched along the x-axis. The ends of the string are secured to the x-axis and the string is initially at rest. Starting at time $t=0$ the section between the ends of the string is released and an external force pointing in the vertical direction is applied to the string. At each point on the string the strength of the force is proportional to the horizontal distance from the left end of the string, with constant of proportionality $c$ that is constant for all $t > 0$. 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 is secured:
Initially at rest:
Initially on $x$-axis: