The ends of a horizontal string of length L are secured at both ends. Initially the string is at rest, stretched straight between its ends. Then, starting when $t \geq 0$, a vertical external force is applied, displacing the string between its two ends. At each point on the strong the strength of the external force is proportional to the horizontal distance from the left end of the strong.
Choose the PDE and boundary/initial conditions that model this scenario.

Partial Differential Equation

A. $\displaystyle\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0, \quad 0<x<a, \quad 0<y<b$

B. $\displaystyle k\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}, \quad 0<x<L, \quad t>0$

C. $\displaystyle\alpha^2\frac{\partial^2 u}{\partial x^2}- \beta x = \frac{\partial^2 u}{\partial t^2}, \quad 0<x<L, \quad t>0$


Boundary/Initial Conditions

A. $\begin{aligned} u(0,t) &= 0 &u(L,t) &= 0, &t > 0\\ u(x,0) & = 0,&\frac{\partial u}{\partial t}\Bigg\vert_{t=0} &= 0, &0 < x < L\\ &&&&\\ &&&&\\ \end{aligned}$
B. $\begin{aligned} u(0,t) &= 0 &u(L,t) &= 0, &t > 0\\ u(x,0) & = x,&& &0 < x < L\\ &&&&\\ &&&&\\ \end{aligned}$
C. $\begin{aligned} \frac{\partial u}{\partial x}\Bigg\vert_{x=0} & = 0, &\frac{\partial u}{\partial x}\Bigg\vert_{x=L} &= 0, &0 < y < b\\ u(x,0) &= 0, &u(x,b) &= x, &0 < x < L\\ &&&&\\ &&&&\\ \end{aligned}$