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Does anyone have ideas about how to solve the following second order Riccati equation?

$$ x'' + \alpha x' = \beta x^2 + \gamma .$$

Thank you.

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First, assuming $x=x(t)$, since there is no $t$ in the equation, we can substitute $x'=p(x)$ and then $x''=x'p'(x)=pp'$. Then the equation turn into a first order equation: $$pp'+\alpha p=\beta x^2+\gamma$$ Does this help?

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This really depends on your variables, if $\beta$ or $\gamma$, but not both are negative and $\alpha=0$ then you can use the linear shift $x=a+u(t)$ where $a=\pm \sqrt{-\frac{\gamma}{\beta}}$ to get the integrable equation

$$u''(t)-a u(t)-\beta a u(t)^2=0$$

Otherwise not sure what you can do, run into this equation a bunch of times, its a pain.

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