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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

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

Leading to

$\displaystyle\begin{align} p_1 &= c_1\\ p_2&=c_2\\ x_1&=2c_1s\\ x_2&=2c_2s\\ z&=n_0s+c_3\end{align}$

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.

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  • $\begingroup$ This computation may be useful $\endgroup$ – Harry49 Nov 9 '18 at 13:52

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