The ends of a string of length L are secured at both ends. Its initial shape is straight but it is vibrating with initial velocity $f(x)=x(L-x)$.
Choose the PDE and boundary/initial conditions that model this scenario.

Partial Differential Equation

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

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

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


Boundary/Initial Conditions

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