What it means that a function depends on $x,y,u,u_x,u_y$?

Question

Definition: A PDE equation is quasilinear if $Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu+G+\Phi(x,y,u,u_x,u_y)=0,$ where $A,B,C,D,E,F,G$ are functions of $x,y,u.$

Is this equation quasilinear? $$(x^2+u^2)u_x-xyu_y=u^3x+y^2$$

The answer is yes it is because $D=(x^2+u^2),E=-xy,F=u^2x, G=y^2$.

However in this pdf http://nptel.ac.in/courses/Webcourse-contents/IIT-%20Guwahati/maths3/module_13/pdenotes.pdf pag. [2] says that $xu_x+yu_y+u^2=0$ is not linear, semilinear nor quasilinear. And I don't understand why, according to definition it's quasilinear.

So maybe I'm not understanding correctly the meaning of function of.. or maybe the pdf file is wrong.

Answer 1

Using the definition in the note:

A PDE is said to be quasilinear if it is linear in its highest derivative. That is, the coefficient of highest order does not depend on any highest order partial derivative.

So the PDE $xu_x + yu_y +u^2 = 0$ IS quasilinear. Indeed in the note it is not claimed that the above is NOT quasilinear. It said that it is "nonlinear".

The $xu_x + yu_y +u^2 = 0$ is not linear because of that $u^2$ term. Thus it is nonlinear. (So it is both quasilinear and nonlinear).