Bessel - Maple Help

Student[ODEs][Solve]

 Bessel
 Solve a 2nd order linear ODE in terms of Bessel functions

 Calling Sequence Bessel(ODE, y(x))

Parameters

 ODE - a second order linear ordinary differential equation y - name; the dependent variable x - name; the independent variable

Description

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

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right):$
 > $\mathrm{ode1}≔{t}^{2}\mathrm{diff}\left(y\left(t\right),t,t\right)+5t\mathrm{diff}\left(y\left(t\right),t\right)+\left({t}^{2}-9\right)y\left(t\right)=0$
 ${\mathrm{ode1}}{≔}{{t}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{5}{}{t}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({{t}}^{{2}}{-}{9}\right){}{y}{}\left({t}\right){=}{0}$ (1)
 > $\mathrm{Bessel}\left(\mathrm{ode1},y\left(t\right)\right)$
 ${y}{}\left({t}\right){=}\frac{{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left(\sqrt{{13}}{,}{t}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left(\sqrt{{13}}{,}{t}\right)}{{{t}}^{{2}}}$ (2)
 > $\mathrm{ode2}≔{t}^{2}\mathrm{diff}\left(y\left(t\right),t,t\right)+t\mathrm{diff}\left(y\left(t\right),t\right)+\left(9{t}^{2}-16\right)y\left(t\right)=0$
 ${\mathrm{ode2}}{≔}{{t}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{t}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({9}{}{{t}}^{{2}}{-}{16}\right){}{y}{}\left({t}\right){=}{0}$ (3)
 > $\mathrm{Bessel}\left(\mathrm{ode2},y\left(t\right)\right)$
 ${y}{}\left({t}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left({4}{,}{3}{}{t}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left({4}{,}{3}{}{t}\right)$ (4)
 > $\mathrm{ode3}≔{t}^{2}\mathrm{diff}\left(y\left(t\right),t,t\right)+5t\mathrm{diff}\left(y\left(t\right),t\right)+\left(9{t}^{2}-25\right)y\left(t\right)=0$
 ${\mathrm{ode3}}{≔}{{t}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{5}{}{t}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}\left({9}{}{{t}}^{{2}}{-}{25}\right){}{y}{}\left({t}\right){=}{0}$ (5)
 > $\mathrm{Bessel}\left(\mathrm{ode3},y\left(t\right)\right)$
 ${y}{}\left({t}\right){=}\frac{{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left(\sqrt{{29}}{,}{3}{}{t}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left(\sqrt{{29}}{,}{3}{}{t}\right)}{{9}{}{{t}}^{{2}}}$ (6)

Compatibility

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