Consider the equation $yu_x-xu_y=0$ for $(x,y)\in \mathbb{R} \times (0,\infty)$ with $u(x,0)=x^2$ as the initial condition.
I just need help solving for the characteristic curve. I have that $$x_t=y, y_t=x, u_t=0$$ but I am not sure how to solve for the characteristic curves when there are three independent variables to solve?
You have, by the method of characteristics:
$$ \frac{dx}{y} = \frac{dy}{-x} = \frac{du}{0}. $$
If you solve the first equality, you will find:
$$x^2+y^2 = C,$$
where $C$ is a constant. On the other hand, the last fraction tells you that $du = 0$ and therefore $u = K$, being $K$ another constant. Put $K$ as a function of $C$ to have the general solution for $u$, which is finally given by:
$$ u = F(x^2+y^2),$$
where $F$ is an arbitrary function of its argument.
Cheers!