Bessel and Modified Bessel ODEs - Maple Programming Help

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Bessel and Modified Bessel ODEs

Description

 • 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);
 ${\mathrm{Bessel_ode}}{≔}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{{n}}^{{2}}{+}{{x}}^{{2}}\right){}{y}{}\left({x}\right)$ (1)
 • 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);
 ${\mathrm{modified_Bessel_ode}}{≔}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\left({{n}}^{{2}}{+}{{x}}^{{2}}\right){}{y}{}\left({x}\right)$ (2)
 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.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (3)
 > $\mathrm{odeadvisor}\left(\mathrm{Bessel_ode}\right)$
 $\left[{\mathrm{_Bessel}}\right]$ (4)
 > $\mathrm{odeadvisor}\left(\mathrm{modified_Bessel_ode}\right)$
 $\left[\left[{\mathrm{_Bessel}}{,}{\mathrm{_modified}}\right]\right]$ (5)

The Bessel ODEs can be solved for in terms of Bessel functions:

 > $\mathrm{dsolve}\left(\mathrm{Bessel_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left({n}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left({n}{,}{x}\right)$ (6)
 > $\mathrm{dsolve}\left(\mathrm{modified_Bessel_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselI}}{}\left({n}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselK}}{}\left({n}{,}{x}\right)$ (7)