# PDE Characteristic Curve Method

## Homework Statement

Solve $$u_x^2+u_y^2=1$$ subject to $$u(x, ax)=1$$

## The Attempt at a Solution

I let $$u_x=p$$ and$$u_y=q$$and$$F=p^2 +q^2 -1=0$$ Then x'=2p, y'=2q, u'=p.2p+q.2q=2, and p'=0=q'. So $$p=p_0, q=q_0$$ are constants. I got $$x'=2p_0, y'=2q_0$$ and integrating the equations for x',y',u' I get $$x=2p_0s+x_0$$ $$y=2q_0s + y_0$$and $$u=2s+u_0$$ I'm not sure when it comes to implementing the initial condition. I think that I can say that $$u_x(x,ax)=p_0=0$$ which gives q=+1 or -1 from the condition of F=0. Can I then fill in $$x_0, ax_0, 1$$ for the initial values of x, y and u respectively? In the end I get an answer $$u=\frac{y}{q_0}-\frac{ax}{q_0}+1$$. Firstly, I'm not sure what to do when $$q_0$$ can have two values. Also, when I fill in my answer into the original PDE I get that a must be zero, which can't be right.

Thanks for any help.