I'm studying a "vibrating circular membrane" problem and I've attached some screenshots of one possible mode of vibration.
The equation for this mode of vibration is $$ \varphi(t,x,y)=\big(3\sin^2(t)x^2+3\cos^2(t)y^2\big)e^{\frac{1}{\log(x^2+y^2)}} $$
for time, $t\in(0,\infty)$ and $x,y\in(-1,1).$ Also $\varphi=0$ on $\partial\Omega.$
However I checked that $\varphi(t,x,y)$ does not satisfy the partial differential equation:
$$ \frac{\partial^2\varphi}{\partial t^2}=c^2\bigg( \frac{\partial^2\varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2} \bigg) $$
So I'm just wondering if $\varphi$ is a solution to a partial differential equation? I'm not sure how to reverse construct the differential equation that $\varphi$ satisfies.
I'd like to characterize the normal modes of this system but if the wave equation isn't satisfied I'm not sure how to proceed.