Question 1: $u_{xx}+u_{xxyy}+u=0$
This is a 4th order linear homogeneous equation since all the terms in the equation are related to u and the operator of the equation $\frac{d^2u}{dx^2}+\frac{d^2u}{dx^2}\frac{d^2u}{dy^2}+1$ is linear.
Question 2: Find the general solution for $u_{xyz}=xy\\
u_{xy}=xyz+f(x,y)\\
u_{x}=\frac{1}{2}xy^2z+F(x,y)+g(x,z)\\
u=\frac{1}{4}x^2y^2z+\hat{F}(x,y)+G(x,z)+h(y,z)$
