Solving PDE w/Fourier: Obtain All Solutions

[walter9459]

Homework Statement

Obtain all solutions of the equation partial ^2 u/partial x^2 - partial u/partial y = u of the form u(x,y)=(A cos alpha x + B sin alphax)f(y) where A, B and alpha are constants. Find a solution of the equation for which u=0 when x=0; u=0 when x = pi, u=x when y=1.

Homework Equations

The solution is u = -2 summuation from n=1 to infinity (((-1)^n)/n)e^((1+n)(1-y)) sin nx.

The Attempt at a Solution

I believe the next step is to use u(x,Y) = X(x)Y(y) so the equation then becomes (1/x) partial ^2 X/partial x^2 - (1/y)partial Y/partial y = u. Then I get lost, can I get some help on how I would solve this problem?

 

[quantumdude]

The Attempt at a Solution

I believe the next step is to use u(x,Y) = X(x)Y(y) so the equation then becomes (1/x) partial ^2 X/partial x^2 - (1/y)partial Y/partial y = u.

No, you forgot to divide the right hand side by u. You should have gotten the following:

$$\frac{X''}{X}-\frac{Y'}{Y}=1$$

Since the first term on the left side depends only on x and the second depends only on y, and since they differ by a constant (namely 1), try setting $X''/X=-\alpha^2$ and $Y'/Y=-\alpha^2-1$. That way you get the desired form of the solution and when you subtract them you get 1.