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!
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2 Answers
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The equation is in separated variable: $$ \frac{dy}{\alpha y + \beta y^2 + \gamma}=dt. $$ Integrate to get $$ \int \frac{dy}{\alpha y + \beta y^2 + \gamma}=t. $$
answered Jul 2, 2013 at 9:39
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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 $$y_1=\frac{-\alpha+\sqrt{\alpha^2-4\beta\gamma}}{2\beta}$$
