At this question the equation $$ u_{xxxx}+4u_{xy}=\tan{x} $$ is being answered as nonlinear.
But as far as i know, the linearity of a DE only depends on the (most) dependent variable. And in this case (and usually in PDEs) that would be the variable $u$. So i think it has to be linear.
See here for example: $$xy \frac{∂^2u}{∂x^2} − e^{x−y} \frac{∂^2u}{∂x∂y} + \frac{∂u}{∂x} − 2xu = \cos(xy) $$
There is the $\cos(xy)$ term which causes the equation to be inhomogeneous, but it will still be linear nonetheless.
Am i missing here something?