<s>[INST] Characteristic method for solving a linear PDE $xyz_x-x^2z_y=-yz$. 
Find the solution of the linear PDE
  $$xyz_x-x^2z_y=-yz$$

Chacteristic equations:
$$\frac{dx}{xy}=\frac{dy}{-x^2}=\frac{dz}{-y}$$
from here, I found the characteristic $x^2+y^2=c_1$ and $lnx+z=c_2$.
since $c_2=f(c_1)$, we have $lnx+z=f(x^2+y^2)$ and solution is $z=f(x^2+y^2)-lnx$.
It is right?
if it is wrong, where am I wrong?
 [/INST] $$\frac{dx}{xy}=\frac{dy}{-x^2}=\frac{dz}{-yz}$$
then with $\dfrac{dx}{xy}=\dfrac{dy}{-x^2}$ we find $x^2+y^2=C_1$ and with $\dfrac{dx}{xy}=\dfrac{dz}{-yz}$ we have $xz=C_2$ thus
$$z=\frac{f(x^2+y^2)}{x}$$
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