# Compute the Characteristics of a PDE

Let $$\displaystyle u_x^2+u_y^2=n_0^2$$ be given, with the initial condition that $$u(x,2x)=1$$ and $$n_0\in\mathbb{R}$$

I want to find a solution using the methods of characteristics. I computed the characteristics to be

\displaystyle \begin{align} \dot{\vec p}&=0\\ \dot z(s)=2\left |\vec p\right |^2&=n_0\\ \dot{\vec x}&=2\vec p\end{align}

\displaystyle\begin{align} p_1 &= c_1\\ p_2&=c_2\\ x_1&=2c_1s\\ x_2&=2c_2s\\ z&=n_0s+c_3\end{align}
My problem is that I can not go on from here. I need an expression for $$u$$ before I can use the BC, but this means solving the equations.