 Chebyshev - Maple Help

Student[ODEs][Solve]

 Chebyshev
 Solve Chebyshev's Equation Calling Sequence Chebyshev(ODE, y(x)) Parameters

 ODE - a Chebyshev equation y - name; the dependent variable x - name; the independent variable Description

 • The Chebyshev(ODE, y(x)) command finds the solution of a Chebyshev equation, which is a linear homogeneous ordinary differential equation of the form:

$\mathrm{ODE}≔\left(-{x}^{2}+1\right)\mathrm{y\text{'}\text{'}}-x\mathrm{y\text{'}}+{p}^{2}y=0$

 • Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one. Examples

 > $\mathrm{with}\left({{\mathrm{Student}}_{\mathrm{ODEs}}}_{\mathrm{Solve}}\right):$
 > $\mathrm{ode1}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{Chebyshev}\left(\mathrm{ode1},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\sqrt{{-}{{x}}^{{2}}{+}{1}}{+}{\mathrm{_C2}}{}{x}$ (2)
 > $\mathrm{ode2}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+4y\left(x\right)=0$
 ${\mathrm{ode2}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}{y}{}\left({x}\right){=}{0}$ (3)
 > $\mathrm{Chebyshev}\left(\mathrm{ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{x}{}\sqrt{{-}{{x}}^{{2}}{+}{1}}{+}{\mathrm{_C2}}{}\left({2}{}{{x}}^{{2}}{-}{1}\right)$ (4)
 > $\mathrm{ode3}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+9y\left(x\right)=0$
 ${\mathrm{ode3}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{9}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{Chebyshev}\left(\mathrm{ode3},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%sin}}{}\left({3}{}{\mathrm{arccos}}{}\left({x}\right)\right)\right]\right){+}{\mathrm{_C2}}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%cos}}{}\left({3}{}{\mathrm{arccos}}{}\left({x}\right)\right)\right]\right)$ (6)
 > $\mathrm{ode4}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-25y\left(x\right)=0$
 ${\mathrm{ode4}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{25}{}{y}{}\left({x}\right){=}{0}$ (7)
 > $\mathrm{Chebyshev}\left(\mathrm{ode4},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{5}{}{\mathrm{arccos}}{}\left({x}\right)}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{5}{}{\mathrm{arccos}}{}\left({x}\right)}$ (8) Compatibility

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