Originally Posted by

**Jose27** Ok, so I've been trying to solve this for a while now and I don't seem to get anywhere. Hope someone can point in the right direction.

We have the PDE

$\displaystyle x(y^2+u)u_x-y(x^2+u)u_y=(x^2-y^2)u$

where $\displaystyle x,y$ are independent variables and $\displaystyle u=u(x,y)$.

So, I know that the equations for the characteristics in this case are

$\displaystyle x_t=x(y^2+u)$

$\displaystyle y_t=-y(x^2+u)$

$\displaystyle u_t=(x^2-y^2)u$

but, I don't see how I can obtain $\displaystyle t,s$ from here, I've tried playing with the coefficients in the original, isolating $\displaystyle u$ or $\displaystyle u_x$ but that gets me nowhere. I'm thinking of proposing the curve $\displaystyle x^2-y^2=c$ as a characteristic, but I haven't checked if it is. So, if someone could give any hints as to how to solve this it would be appreciated.