Generic Solution of a Coupled System of 2nd Order PDEs

[derya]

TL;DR Summary

We are looking for the generic solution of this coupled system of 2nd order PDEs.

Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it.

I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C are constants, and (if it helps) A, B > 0.

 

[pasmith]

It's linear, so you can try looking for separable solutions. If you set $u = Ue^{kx + ly}$ and $v = Ve^{kx + ly}$ then you get the following eigenvalue problem for $k$ and $l$:
$$k^4(1 + B^2C) + k^2 l^2 (2 + C(A^2 + B^2)) + l^4(1 + A^2C) = 0.$$

EDIT: This factorises as $$(k^2 + l^2)(k^2(1 + CB^2) + l^2(1 + CA^2)) = 0.$$

 

[wrobel]

Laplace transform; Fourier transform...