Quasilinear PDE

[NARUTO]

Hi,

 

I'm trying to solve a PDE: u_x + u² u_y = x using the method of characteristics using the following condition u(0,y) = ay where a is constant. So far, I obtained this system of ODE:

 

dx/dt = 1 (1)

 

dy/dt = u² (2)

 

du/dt = x (3)

 

(1) => x = t

(3) => u= 1/2 x² + a s

 

(2) => dy/dt = (1/2 x² + a s)²

 

Now my problem is that when trying to solve (2) (which I manage to do, y= 1/5 t⁵ + 1/3 t³ as + (as)² t + s), I get a second order equation is s, when I tried to solved I found a delta which I don't know its sign and therefore I wasn't able to express s as a function of x and y.

 

Any help is appreciated.

Many thanks.