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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right):$
 > $\mathrm{ode1}≔{x}^{3}\mathrm{diff}\left(y\left(x\right),x,x,x\right)+3{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)-6x\mathrm{diff}\left(y\left(x\right),x\right)-6y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}{{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{IC}≔\left[\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right),x,x\right),x=1\right)=-1,\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right),x\right),x=1\right)=1,y\left(1\right)=2\right]$
 ${\mathrm{IC}}{≔}\left[\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}{=}{-1}{,}\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{1}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}}{\left\{{x}{=}{1}\right\}}{=}{1}{,}{y}{}\left({1}\right){=}{2}\right]$ (2)
 > $\mathrm{HighOrder}\left(\mathrm{ode1},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{3}{}{\mathrm{_C2}}{}{{x}}^{{3}}{-}\frac{{\mathrm{_C1}}}{{x}}{-}\frac{{2}{}{\mathrm{_C3}}}{{{x}}^{{2}}}$ (3)
 > $\mathrm{HighOrder}\left(\mathrm{ode1},y\left(x\right),\mathrm{ICs}=\mathrm{IC}\right)$
 ${y}{}\left({x}\right){=}\frac{{7}{}{{x}}^{{3}}}{{20}}{+}\frac{{13}}{{4}{}{x}}{-}\frac{{8}}{{5}{}{{x}}^{{2}}}$ (4)
 > $\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right),x,x,x\right)+3\mathrm{diff}\left(y\left(x\right),x,x\right)+4\mathrm{diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{3}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{HighOrder}\left(\mathrm{ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{-}{{ⅇ}}^{{-}{x}}{}\left(\left({\mathrm{_C2}}{+}{\mathrm{_C3}}\right){}{\mathrm{sin}}{}\left({x}\right){+}\left({\mathrm{_C2}}{-}{\mathrm{_C3}}\right){}{\mathrm{cos}}{}\left({x}\right){+}{\mathrm{_C1}}\right)$ (6)
 > $\mathrm{HighOrder}\left(\mathrm{ode2},y\left(x\right),\mathrm{ICs}=\mathrm{IC}\right)$
 ${y}{}\left({x}\right){=}{-}{{ⅇ}}^{{-}{x}}{}\left(\left(\frac{{3}{}{\mathrm{sin}}{}\left({1}\right)}{{{ⅇ}}^{{-1}}{}\left({{\mathrm{cos}}{}\left({1}\right)}^{{2}}{+}{{\mathrm{sin}}{}\left({1}\right)}^{{2}}\right)}{-}\frac{{3}{}{\mathrm{cos}}{}\left({1}\right)}{{{ⅇ}}^{{-1}}{}\left({{\mathrm{cos}}{}\left({1}\right)}^{{2}}{+}{{\mathrm{sin}}{}\left({1}\right)}^{{2}}\right)}\right){}{\mathrm{sin}}{}\left({x}\right){+}\left(\frac{{3}{}{\mathrm{sin}}{}\left({1}\right)}{{{ⅇ}}^{{-1}}{}\left({{\mathrm{cos}}{}\left({1}\right)}^{{2}}{+}{{\mathrm{sin}}{}\left({1}\right)}^{{2}}\right)}{+}\frac{{3}{}{\mathrm{cos}}{}\left({1}\right)}{{{ⅇ}}^{{-1}}{}\left({{\mathrm{cos}}{}\left({1}\right)}^{{2}}{+}{{\mathrm{sin}}{}\left({1}\right)}^{{2}}\right)}\right){}{\mathrm{cos}}{}\left({x}\right){-}\frac{{5}}{{{ⅇ}}^{{-1}}}\right)$ (7)

Compatibility

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