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how solve this second-order nonlinear equation

enter image description here

my work I know that we first convert differential X'' to D^2 and try to get the general solution so

suppose D^2 = X''

then

D^2 + (delta)*D+x(x^2-x) =0

i'm stuck in this step , I need just first push to solve the wholly equation

thanks

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  • $\begingroup$ I doubt there is a closed form solution at all. $\endgroup$ – Ian Aug 20 '17 at 16:17
  • $\begingroup$ what should I do now ? , I need to solve this equation $\endgroup$ – user155971 Aug 20 '17 at 16:18
  • $\begingroup$ Are you sure the question wants a closed form solution? Perhaps instead they want some kind of perturbative expansion or qualitative analysis. $\endgroup$ – Ian Aug 20 '17 at 16:18
  • $\begingroup$ yes, it just want to write the equation as system of two first-order $\endgroup$ – user155971 Aug 20 '17 at 16:21
  • $\begingroup$ you can try a numerical solution $\endgroup$ – Dr. Sonnhard Graubner Aug 20 '17 at 16:21
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The equation

$x'' + \delta x' -x + x^3 = \gamma \cos \omega t \tag 1$

is an example of Duffing's equation. No closed form solution is known, and I doubt it has one in terms of elementary functions. The solutions exhibit many important and engaging phenomena, ranging from bounded oscillation to chaos, depending on the parameter values. Check out the linked citing; it's worth reading.

Meanwhile, if what is wanted here is to simply cast (1) into first-order form, we can exploit the standard procedure of setting

$v = x', \tag 2$

so that

$v' = x'', \tag 3$

and then (1) may be written

$v' + \delta v - x + x^3 = \gamma \cos \omega t, \tag 4$

or

$v' = - \delta v + x - x^3 + \gamma \cos \omega t, \tag 5$

which together with (2) forms the first order system

$x' = v, \tag 6$

$v' = - \delta v + x - x^3 + \gamma \cos \omega t \tag 7$

in the two variables $x$ and $v$.

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