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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!

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    $\begingroup$ 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 $G$. $\endgroup$ – Jason Knapp Jul 25 '14 at 17:25
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    $\begingroup$ or he is just missing the exponent in the solution. $\endgroup$ – Chinny84 Jul 25 '14 at 17:28
  • $\begingroup$ Also a possibility :) $\endgroup$ – Jason Knapp Jul 25 '14 at 17:29
  • $\begingroup$ It is missing the squared in the solution which I have fixed $\endgroup$ – Leo Spencer Jul 25 '14 at 17:32
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Hint: $\dfrac{d}{dt}\dfrac{1}{S(t)} = -\dfrac{dS/dt}{S(t)^2}$. What does the Riccati equation imply about the RHS?

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    $\begingroup$ I can't resist pointing out the following irony in me being the first to give an answer: The WKB method in quantum mechanics is premised on the ability to convert the Schrodinger equation to a Riccati equation. And the WKB method is a semiclassical method... $\endgroup$ – Semiclassical Jul 25 '14 at 17:38
  • $\begingroup$ Yeah this was ridiculously easy. As I said it had been a while since I have done DEs so I had a brain freeze. $\endgroup$ – Leo Spencer Jul 25 '14 at 17:58

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