Determining that the singularity of a differential equation is a pole only.

Question

The equation is: $$ \frac{\mathrm{d}w}{\mathrm{d}z} = w-w^2 $$ where $w(z)$ is a complex function. To solve, I separated variables and then used partial fraction decomposition to obtain: $$ \frac{\mathrm{d}w}{w(1-w)} =\mathrm{d}z $$ $$ \frac{\mathrm{d}w}{w}-\frac{\mathrm{d}w}{1-w}=\mathrm{d}z $$ Integrating both sides and combining $\ln$ terms, we have: $$ \frac{w}{1-w}=z-z_0 $$ $$ w = \frac{Ce^z}{1+Ce^z} \quad\quad\text{where}\quad C=e^{-z_0} $$ Now my confusion is how to use this to solution to prove all the singularities are poles. The singularities occur at $$ 1+Ce^z=0 \implies Ce^z=-1 =e^{z-z_0} = -1 $$ $$ \implies z-z_0 = (2k+1)\pi i $$ So then it would be an essential singularity not a pole - right? Any insight would be appreciated.