Halm - Maple Help

Halm ODEs

Description

 • The general form of the Halm ODE is given by the following:
 > Halm_ode := (1+x^2)^2*diff(y(x),x,x)+lambda*y(x) = 0;
 ${\mathrm{Halm_ode}}{≔}{\left({{x}}^{{2}}{+}{1}\right)}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{\mathrm{\lambda }}{}{y}{}\left({x}\right){=}{0}$ (1)
 See Hille, "Lectures on Ordinary Differential Equations", p. 357. The solution to this ODE can be expressed in terms of the hypergeometric function; see hypergeom.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Halm_ode}\right)$
 $\left[{\mathrm{_Halm}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Halm_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\sqrt{{{x}}^{{2}}{+}{1}}{}{\left(\frac{{x}{+}{I}}{{-}{x}{+}{I}}\right)}^{\frac{\sqrt{{\mathrm{\lambda }}{+}{1}}}{{2}}}{+}{\mathrm{_C2}}{}\sqrt{{{x}}^{{2}}{+}{1}}{}{\left(\frac{{x}{+}{I}}{{-}{x}{+}{I}}\right)}^{{-}\frac{\sqrt{{\mathrm{\lambda }}{+}{1}}}{{2}}}$ (4)