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;

$\mathrm{Halm\_ode}≔{\left(x^2+1\right)}^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\mathrm{\lambda }y\left(x\right)=0$

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.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Halm\_ode}\right)$

$\left[\mathrm{\_Halm}\right]$

$\mathrm{dsolve}\left(\mathrm{Halm\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\sqrt{x^2+1}{\left(\frac{x+\mathrm{I}}{-x+\mathrm{I}}\right)}^{\frac{\sqrt{\mathrm{\lambda }+1}}{2}}+\mathrm{c\_\_2}\sqrt{x^2+1}{\left(\frac{x+\mathrm{I}}{-x+\mathrm{I}}\right)}^{-\frac{\sqrt{\mathrm{\lambda }+1}}{2}}$