HighOrder - Maple Help

Student[ODEs][Solve]

 HighOrder
 Solve a high order ODE

 Calling Sequence HighOrder(ODE, y(x))

Parameters

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

Description

 • The HighOrder(ODE, y(x)) command finds the solution of a high order ODE, i.e. where the order is greater than 2.
 • 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

 > $\mathrm{with}\left({{\mathrm{Student}}_{\mathrm{ODEs}}}_{\mathrm{Solve}}\right):$
 > $\mathrm{ode1}≔{x}^{3}\left(\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)\right)+3{x}^{2}\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-6x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-6y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}{{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{IC}≔\left[\genfrac{}{}{0}{}{\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)}{\phantom{x=1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)}}{x=1}=-1,\genfrac{}{}{0}{}{\frac{ⅆ}{ⅆx}y\left(x\right)}{\phantom{x=1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{ⅆ}{ⅆx}y\left(x\right)}}{x=1}=1,y\left(1\right)=2\right]$
 ${\mathrm{IC}}{≔}\left[\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}{=}{-1}{,}\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}{=}{1}{,}{y}{}\left({1}\right){=}{2}\right]$ (2)
 > $\mathrm{HighOrder}\left(\mathrm{ode1},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}\frac{{4}{}{\mathrm{_C3}}{}{{x}}^{{5}}{+}{36}{}{\mathrm{_C2}}{}{x}{+}{9}{}{\mathrm{_C1}}}{{36}{}{{x}}^{{2}}}$ (3)
 > $\mathrm{HighOrder}\left(\mathrm{ode1},y\left(x\right),\mathrm{ICs}=\mathrm{IC}\right)$
 ${y}{}\left({x}\right){=}\frac{{7}{}{{x}}^{{5}}{+}{65}{}{x}{-}{32}}{{20}{}{{x}}^{{2}}}$ (4)
 > $\mathrm{ode2}≔\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)+3\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+4\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+2y\left(x\right)=0$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{3}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{HighOrder}\left(\mathrm{ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{-}\frac{{{ⅇ}}^{{-}{x}}{}\left({\mathrm{_C2}}{}{\mathrm{sin}}{}\left({x}\right){+}{\mathrm{_C3}}{}{\mathrm{cos}}{}\left({x}\right){-}{2}{}{\mathrm{_C1}}\right)}{{2}}$ (6)
 > $\mathrm{HighOrder}\left(\mathrm{ode2},y\left(x\right),\mathrm{ICs}=\mathrm{IC}\right)$
 ${y}{}\left({x}\right){=}{-}{3}{}\left(\left({\mathrm{cos}}{}\left({x}\right){-}{\mathrm{sin}}{}\left({x}\right)\right){}{\mathrm{cos}}{}\left({1}\right){+}{\mathrm{cos}}{}\left({x}\right){}{\mathrm{sin}}{}\left({1}\right){+}{\mathrm{sin}}{}\left({x}\right){}{\mathrm{sin}}{}\left({1}\right){-}\frac{{5}}{{3}}\right){}{{ⅇ}}^{{1}{-}{x}}$ (7)

Compatibility

 • The Student[ODEs][Solve][HighOrder] command was introduced in Maple 2021.
 • The Student[ODEs][Solve][HighOrder] command was updated in Maple 2022.
 • The output option was introduced in Maple 2022.