# First order non-linear ODE: $y' = \alpha y + \beta y^2 + \gamma$

How would I solve a first order non-linear ODE of the general form:

$\frac{d y}{d t} = \alpha y + \beta y^2 + \gamma$

where $\alpha$, $\beta$ and $\gamma$ are constant

Thanks!

The equation is in separated variable: $$\frac{dy}{\alpha y + \beta y^2 + \gamma}=dt.$$ Integrate to get $$\int \frac{dy}{\alpha y + \beta y^2 + \gamma}=t.$$
Another hint is as follows. We know the below OE as Ricatti's equation: $$y'=P(x)+Q(x)y+R(x)y^2,~~~(\star)$$ The method says if $y_1$ is a known particular solution of $(\star)$ then the following linear equaton will reduced $(\star)$ to $$w'+(Q-2y_1R)w=-R$$ where $w=u^{-1}$. Once you get $w$ and then $u$ so our family of solutions of $(\star)$ is: $$y=y_1+u$$ Here we find $P(x)=\gamma,~~Q(x)=\alpha,~~R(x)=\beta$ which are all constants. And it is easily verified that a particular solution of your OE is $$y_1=\frac{-\alpha+\sqrt{\alpha^2-4\beta\gamma}}{2\beta}$$
• $\overset{\small\color{blue}{\bf \diamondsuit\diamondsuit}}{\Large\color{red}\smile}$ Commented Jul 3, 2013 at 0:42
• @amWhy: $\overset{\color{red}{\bf \star\star}}{\Large\color{red}\smile}$ and thanks Commented Jul 3, 2013 at 4:07