Let:\begin{equation*} \frac{\partial^4 u}{\partial x^4}+2\frac{\partial^4 u}{\partial x^2y^2}+\frac{\partial^4 u}{\partial y^4}=\frac{1}{c}\frac{\partial ^2 u}{\partial t^2} \end{equation*}
with the boundary conditions: $u(x,0,t)=u(x,L,t)=u(0,y,t)=u(W,y,t)=0$ and $t>0$
When the mixed derivative isn't there you an solve this by noting that both sides equal a constant, splitting them up, and solving. This gets you a sinusoidal equation that describes how a sheet would move in space and time.
When the mixed derivative term is here however, you cannot solve it like this. I have tried following wikipedia's method for solving mixed PDEs where you end up with 2 different equations in the case of either $X''/X$ or $Y''/Y$ being equal to a constant to obtain either:
$Y^{(4)}-2\lambda_{1}Y''/Y=1/c^{2}T''/T+K_{1}=A$
or
$X^{(4)}-2\lambda_{2}X''/X=1/c^{2}T''/T+K_{1}=A$
I tried to solve these but end up with two sine waves that are completely seperate from each other (for the cases of either part of the partial differential being equal to zero), which seems wrong, seeing as this is the equation for a plate. Could someone please show me how to solve with this problem?