# Laplace's Equation Boundary Problem

1. Aug 8, 2007

### sqrt(-1)

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

I have a two part question, the first part involves solving Laplace's equation

$$u_{xx} + u_{yy} = 0$$

for the boundary conditions
$$u_x(0,y) = u_x(2,y) = 0$$

$$u(x,0) = 0$$

$$u(x,1) = \sin(\pi x)$$
for
$$0 < x < 2, 0 < y < 1$$.

The second part now states a new boundary problem for the same equation, involving the square area defined on $$0 < x < 1, 0 < y < 1$$. This time we have
$$u_x(0,y) = u(1,y) = 0$$

$$u(x,0) = 0$$

$$u(x,1) = 2\sin(\pi x)$$
The question asks me to use the solution from the first boundary problem to solve this problem directly (using a theorem/principle).

2. Relevant equations

3. The attempt at a solution

I have solved the first part using the standard method of separation of variables but I'm rather puzzled as to what this mystery theorem/principle is that can allow me to take my solution from the first problem and apply it directly to the second boundary problem

2. Aug 8, 2007

### HallsofIvy

Staff Emeritus
Since the first problem has x between 0 and 2 and the second has x between 0 and 1, what would happen if you replaced the "x" in the first solution by "2x"?