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Could any one tell me what type of functions may be the solution of this nonlinear differential equation?

$\sigma''+2\sigma+\sigma^2+\sigma^3=k$

Thank you.

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Your question is somehow vague. Can you define "type of function"? –  Siminore Sep 21 '12 at 7:43

1 Answer 1

up vote 4 down vote accepted

Multiply by $\sigma'$ and integrate to obtain

$$ \frac12\sigma'^2+\sigma^2+\frac13\sigma^3+\frac14\sigma^4=k\sigma+C\;. $$

Then

$$ \sigma'=\sqrt{2\left(k\sigma+C-\sigma^2-\frac13\sigma^3-\frac14\sigma^4\right)}\;, $$

so

$$ \int\frac{\mathrm d\sigma}{\sqrt{2\left(k\sigma+C-\sigma^2-\frac13\sigma^3-\frac14\sigma^4\right)}}=x+D\;. $$

Wolfram|Alpha expresses this in terms of an elliptic integral.

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I am not getting anything while clicking the link you have given, could you tell me some $\frac{Ax+B}{Cx+D}$ will be a solution for this one? A guess –  El Angel Exterminador Sep 21 '12 at 8:10
    
I suggest to read up on elliptic integrals on Wikipedia. They don't have solutions in terms of elementary functions. What happens when you click on the link? Did you try waiting for a bit? –  joriki Sep 21 '12 at 8:12

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