Partial Differential Equation with initial conditions

[alex12]

Hello! This is my first post to this excellent forum! I would like some help with this exercise:

$u_{xx} (x,y) + u_{yy} (x,y) = 0$, with $0 < x < 2 \pi$, $0 < y < 4 \pi$
$u_x (0,y) = 0, \, u_x(2 \pi, y) = 0, \, 0< y < 4 \pi$
$u(x,0) = a \cos(2x), \, u(x, 4 \pi) = a \cos^3(x), \, 0<x<2\pi$

I think that the first step is to set $u(x,y) = X(x) Y(y)$, from which the first equation becomes $X''(x) Y(y) + X(x) Y''(y) = 0$. And by dividing with $X(x) Y(y)$ with obtain $\frac{X''(x)}{X(x)} = \frac{Y''(y)}{Y(y)} = - \lambda$

Now, be obtain two ordinary PDEs, $X''(x) + \lambda X(x) = 0$, and $Y''(y) - \lambda Y(y) = 0$.

I don't know how to continue from now one. Especially what to do with the non-homogeneous initial conditions.

Thank you very much in advance!

 

[UltrafastPED]

Without analyzing the equations in detail, consider:

(a) λ is a constant (parameter) shared by both equations;
(b) the ODE's can be solved in the "ordinary" way; don't stop until you have the general solutions for both!
(c) the boundary conditions will not be applied until you have found the general solutions; you may end up having to "patch together" pieces drawn from the two sets of general solutions which are continuous where the patches join.