I am studying PDEs and have the following (seemingly simple) problem:
Find a surface that passes through the curve $x^2+y^2=z=1$ and is orthogonal to the family of surfaces $z(x+y)=c(3z+1)\qquad(c\in\Bbb R)$
After writing down the orthogonality condition (assuming my calculations are correct $(*)$), this yields the following equation: $u(3u+1)(u_x+u_y)-x-y=0$
We usually solve such equations by using the method of characteristics, which tells us (using assumption $(*)$ again) to solve the following characteristic system:
$\begin{align}\dot x=&u(3u+1)\\\dot y=&u(3u+1)\\\dot u =&x+y\end{align}$
Differentiating the last equation of this system with respect to $t$ gives us $\ddot u=\dot x+\dot y$, which using the first two equations gives us $\ddot u = 2u(3u+1)$
After staring at this equation for some time, I decided to ask Wolfram|Alpha. The result seems pretty ugly, so the following questions arise:
Did I make a mistake/am I missing something? Is my approach correct? How do I proceed?
Thanks.