How to distinguish linear differential equations from nonlinear ones?
I know, that e.g.:
$$
px^2+qy^2 = z^3
$$
is linear, but what can I say about the following P.D.E.
$$
p+\log q=z^2
$$
Why?
Here $p=\dfrac{\partial z}{\partial x}, q=\dfrac{\partial z}{\partial y}$
Definition: A P.D.E. is called a Linear Partial Differential Equation if all the derivatives in it are of the first degree.
First of all, the definition you gave is not widely accepted one. PDE is linear if it's reduced form : $$f(x_1,\cdots,x_n,u,u_{x_1},\cdots,u_{x_n},u_{x_1x_1},\cdots)=0$$ is linear function of $u$ and all of it's partial derivatives, i.e. $u,u_{x_1},u_{x_2},\cdots$. So here, the examples you gave are not linear, since the first term of $$-z^3+z_xx^2+z_y y^2=0$$ and $$-z^2+z_z+\log z_y=0$$ are not first order. (in third-order, and second-order, respectively.) However, if you want to use your definition, the first example is ineed linear, (in your definition) and second one is not linear since it's third term is not in first order of $z_y$.
