The Bessel(ODE, y(x)) command finds the solution of a second order linear ODE in terms of Bessel functions.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right)\:$

$\mathrm{ode1}≔t^2\mathrm{diff}\left(y\left(t\right)\,t\,t\right)+5t\mathrm{diff}\left(y\left(t\right)\,t\right)+\left(t^2-9\right)y\left(t\right)=0$

$\mathrm{ode1}≔t^2\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)+5t\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+\left(t^2-9\right)y\left(t\right)=0$

$\mathrm{Bessel}\left(\mathrm{ode1}\,y\left(t\right)\right)$

$y\left(t\right)=\frac{\mathrm{\_C1}\mathrm{BesselJ}\left(\sqrt{13}\,t\right)+\mathrm{\_C2}\mathrm{BesselY}\left(\sqrt{13}\,t\right)}{t^2}$

$\mathrm{ode2}≔t^2\mathrm{diff}\left(y\left(t\right)\,t\,t\right)+t\mathrm{diff}\left(y\left(t\right)\,t\right)+\left(9t^2-16\right)y\left(t\right)=0$

$\mathrm{ode2}≔t^2\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)+t\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+\left(9t^2-16\right)y\left(t\right)=0$

$\mathrm{Bessel}\left(\mathrm{ode2}\,y\left(t\right)\right)$

$y\left(t\right)=\mathrm{\_C1}\mathrm{BesselJ}\left(4\,3t\right)+\mathrm{\_C2}\mathrm{BesselY}\left(4\,3t\right)$

$\mathrm{ode3}≔t^2\mathrm{diff}\left(y\left(t\right)\,t\,t\right)+5t\mathrm{diff}\left(y\left(t\right)\,t\right)+\left(9t^2-25\right)y\left(t\right)=0$

$\mathrm{ode3}≔t^2\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)+5t\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)+\left(9t^2-25\right)y\left(t\right)=0$

$\mathrm{Bessel}\left(\mathrm{ode3}\,y\left(t\right)\right)$

$y\left(t\right)=\frac{\mathrm{\_C1}\mathrm{BesselJ}\left(\sqrt{29}\,3t\right)+\mathrm{\_C2}\mathrm{BesselY}\left(\sqrt{29}\,3t\right)}{9t^2}$

The Student[ODEs][Solve][Bessel] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.