Can anyone help me prove this?
$z(x_1,x_2)=c$ is a characteristic of
$A\frac{\partial^2u}{\partial x_1^2}+B\frac{\partial^2u}{\partial x_1x_2}+C\frac{\partial^2u}{\partial x_2^2}+D\frac{\partial u}{\partial x_1}+E\frac{\partial u}{\partial x_2}+Fu+G=0$
that is, $z$ is a solution of the characteristic equation
$A(z_{x_1})^2+Bz_{x_1}z_{x_2}+C(z_{x_2})^2=0$
$\Leftrightarrow z$ is a solution of
$A(dx_2)^2-B(dx_1)(dx_2)+C(dx_1)^2=0$
or equivalently $A(y')^2-By'+C=0$
Thanks for any help.
Hint: Use the Implicit Function Theorem, i.e. $F(x,y) = 0 \implies \frac{\partial y}{\partial x} = -\frac{F_x}{F_y}$