The general form of the Duffing ODE is given by:

Duffing_ode := diff(y(x),x,x)+y(x)+epsilon*y(x)^3 = 0;

$\mathrm{Duffing\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+y\left(x\right)+\mathrm{\epsilon }{y\left(x\right)}^3=0$

See Bender and Orszag, "Advanced Mathematical Models for Scientists and Engineers", p. 547. The solution of this type of ODE can be expressed in terms of elliptic integrals, as follows:

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Duffing\_ode}\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_missing\_x}\right]\,\mathrm{\_Duffing}\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_x\_y1}\right]\right]$

$\mathrm{dsolve}\left(\mathrm{Duffing\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_2}\mathrm{JacobiSN}\left(\left(\frac{\sqrt{2\mathrm{\epsilon }+4}x}{2}+\mathrm{c\_\_1}\right)\sqrt{-\frac{2}{{\mathrm{c\_\_2}}^2\mathrm{\epsilon }-\mathrm{\epsilon }-2}}\,\frac{\mathrm{c\_\_2}\sqrt{-\left(\mathrm{\epsilon }+2\right)\mathrm{\epsilon }}}{\mathrm{\epsilon }+2}\right)\sqrt{-\frac{2}{{\mathrm{c\_\_2}}^2\mathrm{\epsilon }-\mathrm{\epsilon }-2}}$