Duffing ODEs - Maple Help

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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;
 ${\mathrm{Duffing_ode}}{:=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){+}{y}{}\left({x}\right){+}{\mathrm{ε}}{}{{y}{}\left({x}\right)}^{{3}}{=}{0}$ (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)
 > $\mathrm{dsolve}\left(\mathrm{Duffing_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C2}}{}{\mathrm{JacobiSN}}{}\left(\left(\frac{{1}}{{2}}{}\sqrt{{2}{}{\mathrm{ε}}{+}{4}}{}{x}{+}{\mathrm{_C1}}\right){}\sqrt{{-}\frac{{2}}{{{\mathrm{_C2}}}^{{2}}{}{\mathrm{ε}}{-}{\mathrm{ε}}{-}{2}}}{,}\frac{{\mathrm{_C2}}{}\sqrt{{-}\left({\mathrm{ε}}{+}{2}\right){}{\mathrm{ε}}}}{{\mathrm{ε}}{+}{2}}\right){}\sqrt{{-}\frac{{2}}{{{\mathrm{_C2}}}^{{2}}{}{\mathrm{ε}}{-}{\mathrm{ε}}{-}{2}}}$ (4)