I need to be able to solve the equation for different values of $a$ and $b$.$$\frac{\partial F}{\partial t}=aF^2-(a+b)F+b$$
For $a=b=1$ this becomes a separable Riccati equation which can be easily solved by completing the square.
However I am struggling with the general case, is this even possible? I know I can use wolfram to find solutions for explicit cases, but is there a general solution?
As $$ ax^2-(a+b)x+b=(x-1)(ax-b) $$ you get $$ \int dt = \int\frac{d(aF)}{(aF-a)(aF-b)} $$ which can now be solved via partial fraction decomposition.