NARUTO

Hi, I'm trying to solve a PDE: ux + u2 uy = 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.