Laplace's equation on a rectangle with mixed b.c.s

[sarideli18]

Homework Statement

Solve Laplace's equation on the rectangle 0< x< L, 0< y< H with the boundary conditions du/dx(0, y) = 0, du/dx(L, y)=y, du/dy(x, 0)=0, U(x, H)=x.

Homework Equations

The Attempt at a Solution

I would be able to solve it by separation of variables if the last boundary condition were du/dx(x,H)=x. How can I apply the last boundary condition? Does principle of superposition work?

 

[HallsofIvy]

You need a function that has derivative 0 at y= 0 and value x at y= H. The first condition makes me think of a quadratic. Look at $v(x, y)= (x/H^2)y^2$. And then change your function to w(x,y)= u(x,y)- v(x,y).

 

[sarideli18]

Then how can I go further with separation of variables?

 

[sarideli18]

By the way, Can I assume U=X(x)Y(y) and say Y(H)=1 and Xx(L)=1 ?