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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"?2012-09-21

1 Answers 1

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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 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?2012-09-21