A thin rectangular plate coincides with the region defined by $0\leq x\leq a,0\leq y\leq b$. The left end is held at temperature $e^{-y}$, and the right end is held at temperature 100. The bottom of the plate is held at temperature f(x). The top is held at temperature 0. The temperature in the plate is in a steady state.
Choose the PDE and boundary/initial conditions that model this scenario.

Partial Differential Equation

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

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

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


Boundary/Initial Conditions

A. $\begin{aligned} u(0,y) &= e^{-y} &u(a,y) &= 100, & 0<y<b\\ u(x,0) & = f(x),& u(x,b) &=0, &0 < y < b\\ &&&&\\ &&&&\\ \end{aligned}$
B. $\begin{aligned} \frac{\partial u}{\partial x}\Bigg\vert_{x=0} & = e^{-y}, &\frac{\partial u}{\partial x}\Bigg\vert_{x=a} &= 100, &0 < y < b\\ u(x,0) &= f(x), &u(x,b) &= 0, &0 < x < a\\ &&&&\\ &&&&\\ \end{aligned}$
C. $\begin{aligned} u(0,t) &= 0 &u(a,t) &= 0, &t > 0\\ u(x,0) & = f(x),&\frac{\partial u}{\partial t}\Bigg\vert_{t=0} &= 0, &0 < x < a\\ &&&&\\ &&&&\\ \end{aligned}$