Solve PDE $xzu_x+yzu_y-(x^2+y^2)u_z=0$ using characteristics method I was asked to find the characteristics of the following EDP and then to solve it.
$$xzu_x+yzu_y-(x^2+y^2)u_z=0$$
I've reached the following characteristic system $$x'=-\frac{xz}{x^2+y^2}\qquad y'= -\frac{yz}{x^2+y^2}
$$ but i dont know how to continue from this. How does one solve an PDE using the characteristic method?
 A: Multiplying the first equation by $x$ and the second one by $y$ gives
$$
\tfrac12 (x^2)' = -\frac{x^2 z}{x^2 + y^2}, \qquad \tfrac12 (y^2)' = -\frac{y^2 z}{x^2 + y^2} .
$$
The sum leads to $(r^2)' = -2z$ where $r^2 = x^2 + y^2$. Multiplying the first equation by $y$ and the second one by $x$ gives
$$
yx' = -\frac{xy z}{x^2 + y^2}, \qquad xy' = -\frac{xy z}{x^2 + y^2} .
$$
The difference leads to $t' = 0$ where $t = y/x$. The additional equations $u' = 0$ and $z' = 1$ yield $u = c_1$ and $z = s + c_2$, respectively. Thus, integrating the differential equations for $r^2$ and $t$, we find $r^2 = -s(s + 2 c_2) + c_3$ and $t = c_4$. The fact that
$z^2 = -r^2+ c_3 + (c_2)^2$
leads to a general solution of the form $u = F(x^2+y^2+z^2, y/x)$ for some arbitrary $F$.
The above steps suggest that the cylindrical coordinates $(x,y,z) \mapsto (r, \theta, z)$ may be relevant here. Indeed, the PDE rewrites as
$zu_r - r u_z = 0$, which general solution is $u = G(r^2 + z^2, \theta)$ for some arbitrary $G$.
