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

$\xi\equiv u_x$

$\eta \equiv u_y$

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


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


I cannot tell what classification these equations have.

  • $\begingroup$ I think you mean $\xi_y = \eta_x$, not $\xi_x = \eta_y$. $\endgroup$ – Robert Israel May 19 '13 at 20:19
  • $\begingroup$ I've corrected it, I think in my head when I say eta I think of xi looking like "e" $\endgroup$ – shilov May 19 '13 at 20:35

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.


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