# Solution methods for the Riccati equation $\dot P = a P^2 - b P$

I have looked all over the place and it seems I can't find a reference for solution methods to common forms of first order nonlinear ordinary differential equations (ODEs). Currently, I am starting at the following population model

$$\dot P = a P^2 - b P$$

given $$a, b > 0$$, and not really getting anywhere with this. It seems that Wolfram|Alpha cannot parse it so I can't search the solution steps for insight, though maybe there's some special syntax I should be using there.

How would I go about solving this one in particular, and where would I find a compendium of ODE solution methods?

• Did the answer below resolve your issues? If so, you should upvote and/or accept the answer. Regards Oct 3, 2013 at 16:10

Hints:

• The equation is separable.
• Write:

$$\displaystyle \int \frac{dP}{a P^2 - b P} = \int dt$$

Can you take it from here?

Book References

• Nonlinear Ordinary Differential Equations, Jordan and Smith

You might also want to visit Equation World.

• Sure, so $ln(aP^2 - bP) = t + C_1$, then $aP^2 - bP = Ce^t$ which is great except that I still have no idea what P(t) is, and I don't have an initial condition to get rid of the integration constant placeholder C. Even if I'm being particularly clever and noting that $aP^2 - bP = Ce^t$ could be substituted into the initial differential equation, I'm left with $dP/dt = Ce^t$ which becomes $P = C_1 e^t + C_2$ after integration, so I'm even worse off than before. Where am I going wrong here?
– user98131
Oct 2, 2013 at 2:22
• Did you do partial fractions on the LHS? The final result should be $P(t) = \dfrac{b}{a + e^{b(t + c)}}$. Try substituting this result back into the original. What do you get? Regards Oct 2, 2013 at 2:37
• Nice work, @Amzoti! +1 Oct 2, 2013 at 11:28
• I'll be thinking about you! Hope it's nothing serious! My visits are going fine: same ol' same ol'! But I'm striking out on the little time I've had for MSE! Oct 2, 2013 at 12:49