# First order quasilinear Partial Differential Equation methods

What methods, not including the one of characteristic curves and surfaces, can I use to solve first-order quasilinear equations like this one? $$u_x+u_y=u^2$$

• Change of coordinates. – user147263 Feb 17 '15 at 2:21
• @Famous Blue Raincoat: you mean like using $s = x + y$, $t = x - y$; then doesn't $u_x + u_y$ become $u_s$ so we have $u_s = u^2$ etc.? – Robert Lewis Feb 17 '15 at 2:28
• @RobertLewis Yes, that's what I meant. It's characteristics in disguise, but textbooks may treat it as a separate method. – user147263 Feb 17 '15 at 2:29
• @Famous Blue Raincoat: thought so, had the same feel. Thanks. – Robert Lewis Feb 17 '15 at 2:31

## 1 Answer

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{dy}{dt}=1$ , letting $y(0)=y_0$ , we have $y=t+y_0=x+y_0$

$\dfrac{du}{dt}=u^2$ , we have $u(x,y)=\dfrac{1}{f(y_0)-t}=\dfrac{1}{f(y-x)-x}$

• ...not including the one of characteristic curves and surfaces... – themaker Feb 17 '15 at 16:06