The general form of the Bessel ODE is given by the following:

Bessel_ode := x^2*diff(y(x),x,x)+x*diff(y(x),x)+(x^2-n^2)*y(x);

$\mathrm{Bessel\_ode}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-n^2+x^2\right)y\left(x\right)$

The general form of the modified Bessel ODE is given by the following:

modified_Bessel_ode := x^2*diff(y(x),x,x)+x*diff(y(x),x)-(x^2+n^2)*y(x);

$\mathrm{modified\_Bessel\_ode}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-\left(n^2+x^2\right)y\left(x\right)$

where n is an integer. See Abramowitz and Stegun - `Handbook of Mathematical Functions`, section 9.6.1. The solutions for these ODEs are expressed using the Bessel functions in the following examples.

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

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

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

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

$\mathrm{odeadvisor}\left(\mathrm{modified\_Bessel\_ode}\right)$

$\left[\left[\mathrm{\_Bessel}\,\mathrm{\_modified}\right]\right]$

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

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{BesselJ}\left(n\,x\right)+\mathrm{c\_\_2}\mathrm{BesselY}\left(n\,x\right)$

$\mathrm{dsolve}\left(\mathrm{modified\_Bessel\_ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{BesselI}\left(n\,x\right)+\mathrm{c\_\_2}\mathrm{BesselK}\left(n\,x\right)$