How can I progress with finding the solution to this quadratic Riccati equation?

Question

I have the following riccati equation:

$\frac{dF}{dt}=\alpha F^2-(\alpha+\beta)F+\beta$

Where $\alpha$ and $\beta$ are real numbers. I am assuming the best way to solve such an integral is through a separation of variables. This leads me to:

$\int \frac{1}{\alpha F^2-(\alpha+\beta)F+\beta}dF=\int dt$

However I have no idea how to progress any further with the integral on the LHS. Any ideas?

Answer 1

Note that $\alpha F^2 + (\alpha+\beta)F + \beta = (F+1)(\alpha F + \beta)$ and thus by partial fraction decomposition, $$\frac{1}{\alpha F^2 + (\alpha+\beta)F + \beta} = \frac{1}{\beta-\alpha}\frac{1}{F+1} - \frac{\alpha}{\beta-\alpha} \frac{1}{\alpha F + \beta}.$$ Can you solve it now?