Ricatti differential equation solution

I attempting to solve some Riccati differential equations. It has been a while since I have worked with differential equations so I am rusty. I would appreciate if someone would show me how to do the following example from my book which they did not do the work for. Then maybe I should be able to do some on my own.

$dS/dt=-G(t)^2S(t)^2$

The solution is $(1/S(0)+\int_0^t G(r)^2 dr)^{-1}$. I can prove this solution is correct but I cannot find the solution on my own.

Thanks for the help!

• Is it definitely $G(t)^2$? I'm not sure of the context here but that solution looks a lot like separation of variables if it were just $G$. – Jason Knapp Jul 25 '14 at 17:25
• or he is just missing the exponent in the solution. – Chinny84 Jul 25 '14 at 17:28
• Also a possibility :) – Jason Knapp Jul 25 '14 at 17:29
• It is missing the squared in the solution which I have fixed – Leo Spencer Jul 25 '14 at 17:32

Hint: $\dfrac{d}{dt}\dfrac{1}{S(t)} = -\dfrac{dS/dt}{S(t)^2}$. What does the Riccati equation imply about the RHS?