Solving first order PDE: $yz\frac{\partial z}{\partial x} - xz\frac{\partial z}{\partial y} = e^z$

Question

I'm trying to solve this PDE:

$yz\frac{\partial z}{\partial x} - xz\frac{\partial z}{\partial y} = e^z$

I have tried method of characteristics:

$\frac{dx}{ds} = yz\\ \frac{dy}{ds} = -xz\\ \frac{dz}{ds} = e^z$

From first two equations I got $x^2 + y^2 = C_1$ and from the third one $z = -ln(c-s)$. What can I do next to get $C_2$?

Answer 1

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