Duffing - Maple Help

Duffing ODEs

Description

 • The general form of the Duffing ODE is given by:
 > Duffing_ode := diff(y(x),x,x)+y(x)+epsilon*y(x)^3 = 0;
 ${\mathrm{Duffing_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{y}{}\left({x}\right){+}{\mathrm{\epsilon }}{}{{y}{}\left({x}\right)}^{{3}}{=}{0}$ (1)
 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:

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Duffing_ode}\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_missing_x}}\right]{,}{\mathrm{_Duffing}}{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_x_y1}}\right]\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Duffing_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C2}}{}{\mathrm{JacobiSN}}{}\left(\left(\frac{\sqrt{{2}{}{\mathrm{\epsilon }}{+}{4}}{}{x}}{{2}}{+}{\mathrm{_C1}}\right){}\sqrt{{-}\frac{{2}}{{\mathrm{\epsilon }}{}{{\mathrm{_C2}}}^{{2}}{-}{\mathrm{\epsilon }}{-}{2}}}{,}\frac{{\mathrm{_C2}}{}\sqrt{{-}\left({\mathrm{\epsilon }}{+}{2}\right){}{\mathrm{\epsilon }}}}{{\mathrm{\epsilon }}{+}{2}}\right){}\sqrt{{-}\frac{{2}}{{\mathrm{\epsilon }}{}{{\mathrm{_C2}}}^{{2}}{-}{\mathrm{\epsilon }}{-}{2}}}$ (4)