Problem: For equation: $$u=xu_x+yu_y+\frac{1}{2}(u_x^2+u_y^2) $$ find the solution with $$u(x,0)=\frac{1}{2}(1-x^2)$$
Here is what I have so far:
Let $$F(x,y,z,p,q)=z-xp-yq-\frac{1}{2}(p^2+q^2)$$ with \begin{align} p&=u_x,\\ q&=u_y \end{align}
Characteristic Equations:
\begin{align} \frac{\Bbb dx}{\Bbb dt}&= -x-p \\ \frac{\Bbb dy}{\Bbb dt}&=-y-q \\ \frac{\Bbb dz}{\Bbb dt}&=-px-p^2-qy-q^2 \\ \frac{\Bbb dp}{\Bbb dt}&=0 \\ \frac{\Bbb dq}{\Bbb dt}&=0 \end{align} I know that I need to get the Characteristic strips (Solutions of Characteristic Equations) but I am stuck. Please help. Thanks
From last two equations: $$p=a, ~~q=b$$ where $a$, $b$ are arbitrary constants. First three equations can be written as $$dz=pdx+q dy=adx+bdy.$$ Integration gives$$z=ax+by+c$$ You can find $c={a^2+b^2\over 2}$
This is a special case in Charpit's method and the equation is called as Clairaut's equation.