Finding polynomial satistying potential equation and boundary conditions

Question

Can someone help me with this problem? I know that this polynomial is a solution of Poisson's equation.

Answer 1

Let $v(x,y)=A+Bx+Cy+Dx^2+Exy+Fy^2$.

First from $v(0,y)=0$ you have:

$A+Cy+Fy^2=0$ for all $0<y<b$, so $A=C=F=0$.

Next from $v(x,0)=0$:

$A+Bx+Dx^2=0$ for all $0<x<a$, so $A=B=D=0$.

Finally from $v(x,b)=x$:

$A+Bx+Dx^2+Cb+Fb^2+Exb=Exb=x$ for $0<x<a$, so $E=\frac{1}{b}$.

Answer 2

Since $v(0,y) = 0$, we know that $x = 0$ is a root of the polynomial, so $x$ is a factor.

Since $v(x,0) = 0$, we know that $y = 0$ is a root of the polynomial, so $y$ is a factor.

This means that $v(x,y) = Exy$ for some constant $E$. Using $v(x,b) = Ebx = x$ yields $E = \frac{1}{b}$.

Thus, $v(x,y) = \dfrac{1}{b}xy$.