Jacobi - Maple Help

Jacobi ODEs

Description

 • The general form of the Jacobi ODE is given by the following:
 > Jacobi_ode := diff(y(x),x,x)*x*(1-x) = (g-(a+1)*x)*diff(y(x),x)+n*(a+n)*y(x);
 ${\mathrm{Jacobi_ode}}{≔}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{}\left({1}{-}{x}\right){=}\left({g}{-}\left({a}{+}{1}\right){}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{n}{}\left({a}{+}{n}\right){}{y}{}\left({x}\right)$ (1)
 where n is an integer. See Iyanaga and Kawada, "Encyclopedic Dictionary of Mathematics", p. 1480.

Examples

The solution to this type of ODE can be expressed in terms of the hypergeometric function; see hypergeom.

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Jacobi_ode}\right)$
 $\left[{\mathrm{_Jacobi}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Jacobi_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{hypergeom}}{}\left(\left[{-}{1}{-}\frac{{a}}{{2}}{-}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}{,}{-}{1}{-}\frac{{a}}{{2}}{+}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}\right]{,}\left[{-}{g}\right]{,}{x}\right){+}{\mathrm{_C2}}{}{{x}}^{{1}{+}{g}}{}{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{a}}{{2}}{-}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}{+}{g}{,}{-}\frac{{a}}{{2}}{+}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}{+}{g}\right]{,}\left[{2}{+}{g}\right]{,}{x}\right)$ (4)