BySeries - Maple Help
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Student[ODEs][Solve]

 BySeries
 Find a series solution for a linear homogeneous ODE with polynomial coefficients

 Calling Sequence BySeries(ODE, y(x))

Parameters

 ODE - an ordinary differential equation y - name; the dependent variable x - name; the independent variable

Description

 • The BySeries(ODE, y(x)) command finds a particular series solution of a linear homogeneous ODE with polynomial coefficients.
 • Note that the series solution may not represent the complete solution of the given ODE.
 • 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

 > $\mathrm{with}\left({{\mathrm{Student}}_{\mathrm{ODEs}}}_{\mathrm{Solve}}\right):$
 > $\mathrm{ode1}≔\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{BySeries}\left(\mathrm{ode1},y\left(x\right)\right)$
 $\left[{y}{}\left({x}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}{{a}}_{{k}}{}{{x}}^{{k}}{,}{{a}}_{{k}{+}{2}}{=}{-}\frac{{{a}}_{{k}}}{{k}{+}{2}}\right]$ (2)
 > $\mathrm{ode3}≔{x}^{2}\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+{x}^{2}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left({x}^{3}-6\right)y\left(x\right)=0$
 ${\mathrm{ode3}}{≔}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{3}}{-}{6}\right){}{y}{}\left({x}\right){=}{0}$ (3)
 > $\mathrm{BySeries}\left(\mathrm{ode3},y\left(x\right)\right)$
 $\left[{y}{}\left({x}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}{{a}}_{{k}}{}{{x}}^{{k}{+}{3}}{,}{{a}}_{{k}{+}{3}}{=}{-}\frac{{k}{}{{a}}_{{k}{+}{2}}{+}{{a}}_{{k}}{+}{5}{}{{a}}_{{k}{+}{2}}}{\left({k}{+}{8}\right){}\left({k}{+}{3}\right)}{,}{{a}}_{{1}}{=}{-}\frac{{{a}}_{{0}}}{{2}}{,}{{a}}_{{2}}{=}\frac{{{a}}_{{0}}}{{7}}\right]$ (4)
 > $\mathrm{ode4}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+\frac{ⅆ}{ⅆx}y\left(x\right)+{x}^{2}y\left(x\right)=0$
 ${\mathrm{ode4}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{{x}}^{{2}}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{BySeries}\left(\mathrm{ode4},y\left(x\right)\right)$
 $\left[{y}{}\left({x}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}{{a}}_{{k}}{}{{x}}^{{k}}{,}{{a}}_{{k}{+}{4}}{=}{-}\frac{{k}{}{{a}}_{{k}{+}{3}}{+}{{a}}_{{k}}{+}{3}{}{{a}}_{{k}{+}{3}}}{{{k}}^{{2}}{+}{7}{}{k}{+}{12}}{,}{{a}}_{{2}}{=}{-}\frac{{{a}}_{{1}}}{{2}}{,}{{a}}_{{3}}{=}\frac{{{a}}_{{1}}}{{6}}\right]$ (6)
 > $\mathrm{ode5}≔\frac{ⅆ}{ⅆx}\left(\left(-{x}^{2}+1\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)+12y\left(x\right)=0$
 ${\mathrm{ode5}}{≔}{-}{2}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{12}{}{y}{}\left({x}\right){=}{0}$ (7)
 > $\mathrm{BySeries}\left(\mathrm{ode5},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{{a}}_{{0}}{}\left(\frac{{3}}{{2}}{}{x}{-}\frac{{5}}{{2}}{}{{x}}^{{3}}\right)$ (8)

The BySeries command accepts only linear homogeneous ODEs with polynomial (or rational polynomial) coefficients. Attempting to use it for an ODE not of this form results in an error message.

 > $\mathrm{ode6}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)=\mathrm{sin}\left(x\right)y\left(x\right)$
 ${\mathrm{ode6}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{\mathrm{sin}}{}\left({x}\right){}{y}{}\left({x}\right)$ (9)
 > $\mathrm{BySeries}\left(\mathrm{ode6},y\left(x\right)\right)$

Compatibility

 • The Student[ODEs][Solve][BySeries] command was introduced in Maple 2021.
 • For more information on Maple 2021 changes, see Updates in Maple 2021.
 • The Student[ODEs][Solve][BySeries] command was updated in Maple 2022.
 • The output option was introduced in Maple 2022.
 • For more information on Maple 2022 changes, see Updates in Maple 2022.

 See Also