solving differential equations : $ay' + by^2 + cy = u$ solving differential equations: $ay' + by^2 + cy = u$ with $a,b,c$ are positive constants and $u$ is arbitrary constant. 
I can solve this equation: 
$ay' + by^2 + cy = 0$
$\Rightarrow -a\frac{dy}{y(by+c)}=dx $
$\Rightarrow \frac{a}{c}(\frac{bdy}{by+c}-\frac{dy}{y})=dx$
$\Rightarrow ln(\frac{by+c}{y})=\frac{c}{a}x$
$\Rightarrow y = \frac{c}{e^{\frac{c}{a}x}-b}$
But with an arbitrary constant at the righthand side, I have no idea.

This is an equation of the Riccati type. One can set $y=\frac{az'}{bz}$, then
$$
u=ay'+by^2+cy=\frac{a^2z''}{bz}+\frac{caz'}{bz}\\~\\
\iff
a^2z''+acz'-ubz=0
$$
This now is a second order linear DE with constant coefficients, so standard solution methods apply. The characteristic roots are solutions of the quadratic polynomial
$$
0=4a^2\lambda^2+4acλ-4ub=(2aλ-c)^2-(c^2+4ub)
$$
