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.
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.
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.