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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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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! – Jacques Carette Aug 4 '10 at 16:02
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? – Loop Space Aug 4 '10 at 16:20
If possible, please give more information in the title of your question. Titles on MO can be up to 240 characters --- almost two tweets. – Theo Johnson-Freyd Aug 4 '10 at 19:37
This reads like homework. I'm voting to close. I echo Theo's plea for a more descriptive title. – José Figueroa-O'Farrill Aug 4 '10 at 22:31
Hey guys, thanks for your comments, by letting $h=(e^{\rho s}g)^{\frac{1}{1-\gamma}}$ I got the solution for this ! – Lam Aug 5 '10 at 14:43
up vote 0 down vote accepted

This seems to be a Bernoulli differential equation. Please cf. for the solution (in your case $n= \frac{\gamma}{\gamma-1}$).

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