 Complete Elliptic Integral ODEs - Maple Programming Help

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Complete Elliptic Integral ODEs

Description

 • The general forms of the elliptic ODEs are given by the following:
 > elliptic_I_ode := diff(x*(1-x^2)*diff(y(x),x),x)-x*y(x)=0;
 ${\mathrm{elliptic_I_ode}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{2}{}{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{x}{}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}{y}{}\left({x}\right){=}{0}$ (1)
 > elliptic_II_ode := (1-x^2)*diff(x*diff(y(x),x),x)+x*y(x)=0;
 ${\mathrm{elliptic_II_ode}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{x}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)\right){+}{x}{}{y}{}\left({x}\right){=}{0}$ (2)
 See Gradshteyn and Ryzhik, "Tables of Integrals, Series and Products", p. 907. The solution to this type of ODE can be expressed in terms of the EllipticK and EllipticCK functions.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{elliptic_I_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{EllipticK}}{}\left({x}\right){+}{\mathrm{_C2}}{}{\mathrm{EllipticCK}}{}\left({x}\right)$ (4)
 > $\mathrm{dsolve}\left(\mathrm{elliptic_II_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{EllipticE}}{}\left({x}\right){+}{\mathrm{_C2}}{}\left({\mathrm{EllipticCE}}{}\left({x}\right){-}{\mathrm{EllipticCK}}{}\left({x}\right)\right)$ (5)
 > $\mathrm{odeadvisor}\left(\mathrm{elliptic_I_ode}\right)$
 $\left[\left[{\mathrm{_elliptic}}{,}{\mathrm{_class_I}}\right]\right]$ (6)
 > $\mathrm{odeadvisor}\left(\mathrm{elliptic_II_ode}\right)$
 $\left[\left[{\mathrm{_elliptic}}{,}{\mathrm{_class_II}}\right]\right]$ (7)