# Is there any analytical solution of this system of two linear inhomogeneous first-order PDE?

So, I have a system of linear inhomogeneous PDE for a function $u(\rho, z)$: $$u_{,\rho} = \rho \phi_{,z}\\ u_{,z} = -\rho \phi_{,\rho}$$ with $\phi(\rho, z)$ being an axially symmetric solution of Laplace equation in cylindrical polar coordinates: $\phi_{,\rho\rho}+\frac{1}{\rho}\phi_{,\rho}+\phi_{,zz} = 0$.

My question is whether there is any procedure/trick to obtain a solution of this problem analytically.

For example, I've found out that it is possible to solve such an equation without any assumptions about $\phi(\rho, z)$: $$u_{,\rho}\phi_{,z}-u_{,z}\phi_{,\rho} = 0$$ Indeed, $u = Ф(\phi(\rho, z))$

I have tried reducing my problem to a single equation $u_{,\rho}\phi_{,\rho}+u_{,z}\phi_{,z} = 0$, but, unfortunately, do not see any sign of success in such a way. Method of characteristics won't work for general enough $\phi(\rho, z)$.

I will appreciate any kind of help or advice.

• Trying to solve 3 equations and 2 unknowns usually ends badly.
– Paul
Commented Sep 26, 2017 at 1:06
• @Paul it seems to be a well-posed problem, however: it's easy to check that $u_{z \rho} = u_{\rho z}$ and therefore there's nothing bad in the 3rd equation. Do you know a method for general $\phi$? Commented Sep 26, 2017 at 1:37
• I'd try to look for a consistent set of symmetry transformations.
– Paul
Commented Sep 26, 2017 at 1:54
• I edited your question to put the comma into $\phi_{,\rho}$ and $\phi_{,zz}$ in the equation $\phi_{,\rho \rho} + \frac{1}{\rho} \phi_{,\rho} + \phi_{,zz} = 0$. Commented Sep 26, 2017 at 16:27
• @RobertLewis, thank you:) Commented Sep 26, 2017 at 17:28