Gegenbauer - Maple Help

Gegenbauer ODEs

Description

 • The general form of the Gegenbauer ODE is given by the following:
 > Gegenbauer_ode := (x^2-1)*diff(y(x),x,x)-(2*m+3)*x*diff(y(x),x)+lambda*y(x)=0;
 ${\mathrm{Gegenbauer_ode}}{≔}\left({{x}}^{{2}}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\left({2}{}{m}{+}{3}\right){}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{\mathrm{\lambda }}{}{y}{}\left({x}\right){=}{0}$ (1)
 where m is an integer. See Infeld and Hull, "The Factorization Method". The solution of this type of ODE can be expressed in terms of the LegendreQ and LegendreP functions:

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Gegenbauer_ode}\right)$
 $\left[{\mathrm{_Gegenbauer}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Gegenbauer_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\left({{x}}^{{2}}{-}{1}\right)}^{\frac{{5}}{{4}}{+}\frac{{m}}{{2}}}{}{\mathrm{LegendreP}}{}\left(\sqrt{{{m}}^{{2}}{-}{\mathrm{\lambda }}{+}{4}{}{m}{+}{4}}{-}\frac{{1}}{{2}}{,}\frac{{5}}{{2}}{+}{m}{,}{x}\right){+}{\mathrm{_C2}}{}{\left({{x}}^{{2}}{-}{1}\right)}^{\frac{{5}}{{4}}{+}\frac{{m}}{{2}}}{}{\mathrm{LegendreQ}}{}\left(\sqrt{{{m}}^{{2}}{-}{\mathrm{\lambda }}{+}{4}{}{m}{+}{4}}{-}\frac{{1}}{{2}}{,}\frac{{5}}{{2}}{+}{m}{,}{x}\right)$ (4)