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let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function" Assume that $0<\gamma<1$, $v$ is a positive number, and $$\frac{dg}{ds}+(v\gamma) g +(1-\gamma)(e^{\rho s}g)^{\frac{1}{\gamma-1}}g=0$$

Find a closed form for $g$?

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  • $\begingroup$ Up to an implicit algebraic equation, yes. Ask Maple. The answer is large enough that I won't paste it here. But it's not so hard to do even by hand! $\endgroup$
    – Jacques Carette
    Commented Aug 4, 2010 at 16:02
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    $\begingroup$ Please provide some context: why are you interested in this equation? Why do you particularly want a closed form (given that so many ODEs don't have closed forms)? What have you done already to try to find one? $\endgroup$
    – Andrew Stacey
    Commented Aug 4, 2010 at 16:20
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    $\begingroup$ If possible, please give more information in the title of your question. Titles on MO can be up to 240 characters --- almost two tweets. $\endgroup$
    – Theo Johnson-Freyd
    Commented Aug 4, 2010 at 19:37
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    $\begingroup$ This reads like homework. I'm voting to close. I echo Theo's plea for a more descriptive title. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Aug 4, 2010 at 22:31
  • $\begingroup$ Hey guys, thanks for your comments, by letting $h=(e^{\rho s}g)^{\frac{1}{1-\gamma}}$ I got the solution for this ! $\endgroup$
    – Lam
    Commented Aug 5, 2010 at 14:43

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This seems to be a Bernoulli differential equation. Please cf. http://en.wikipedia.org/wiki/Bernoulli_differential_equation for the solution (in your case $n= \frac{\gamma}{\gamma-1}$).

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