Writing a 2nd order PDE as a system of equations

Question

I want to turn this 2nd order equation into a system of first order equations but I am unsure about whether I can get rid of the $u$ or not

$$u_{xy}-u_x+u_y+10u u_{xx}$$

To write this as a system of equations so I can determine whether its semi-linear, quasilinear or nonlinear I thought that

$\xi\equiv u_x$

$\eta \equiv u_y$

$$\xi_y-\xi+\eta+10u \xi_x=0$$

$$\xi_y=\eta_x$$

Form a system of 2 equations.

But $u$ is still in it? Does this say something about whether it is quasi-linear or not? Is $u\equiv u(x,y)$ still? If it is then the equation is in the form

$\xi(x,y)_y+f(x,y)\xi(x,y)_x+g(\xi,\eta)=0$

I cannot tell what classification these equations have.

Answer 1

As far as I can tell, "quasilinear" means "linear in the highest-order derivatives". Here the highest-order derivatives are $u_{xy}$ and $u_{xx}$, and the equation is indeed linear in those. The fact that the coefficient of $u_{xx}$ involves $u$ is no problem for quasilinear. On the other hand, since this coefficient does involve $u$ rather than depending only on the independent variables $x,y$, your equation is not semi-linear.