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}

Choose the PDE and boundary/initial conditions that model this scenario.

Partial Differential Equation

Boundary/Initial Conditions

You can earn partial credit on this problem.