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

1. ### sarideli18

9
1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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?

2. ### HallsofIvy

40,314
Staff Emeritus
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).

3. ### sarideli18

9
Then how can I go further with separation of variables?

4. ### sarideli18

9
By the way, Can I assume U=X(x)Y(y) and say Y(H)=1 and Xx(L)=1 ?