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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right)\:$

$\mathrm{ode1}≔x^3\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3x^2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-6x\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ode1}≔x^3\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+3x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-6x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-6y\left(x\right)=0$

$\mathrm{IC}≔\left[\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\,x\right)\,x=1\right)=-1\,\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x=1\right)=1\,y\left(1\right)=2\right]$

$\mathrm{IC}≔\left[\genfrac{}{}{0ex}{}{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}}{\left\{x=1\right\}}=\mathrm{-1}\,\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\phantom{\left\{x=1\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}}{\left\{x=1\right\}}=1\,y\left(1\right)=2\right]$

$\mathrm{HighOrder}\left(\mathrm{ode1}\,y\left(x\right)\right)$

$y\left(x\right)=\frac{\mathrm{c\_\_1}{x}^5+\mathrm{c\_\_3}x+\mathrm{c\_\_2}}{x^2}$

$\mathrm{HighOrder}\left(\mathrm{ode1}\,y\left(x\right)\,\mathrm{ICs}=\mathrm{IC}\right)$

$y\left(x\right)=\frac{7x^5+65x-32}{20x^2}$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=0$

$\mathrm{ode2}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)+3\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4\frac{\ⅆ}{\ⅆx}y\left(x\right)+2y\left(x\right)=0$

$\mathrm{HighOrder}\left(\mathrm{ode2}\,y\left(x\right)\right)$

$y\left(x\right)={\ⅇ}^{-x}\left(\mathrm{c\_\_1}+\mathrm{c\_\_2}\sin \left(x\right)+\mathrm{c\_\_3}\cos \left(x\right)\right)$

$\mathrm{HighOrder}\left(\mathrm{ode2}\,y\left(x\right)\,\mathrm{ICs}=\mathrm{IC}\right)$

$y\left(x\right)=-{\ⅇ}^{1-x}\left(-5+3\left(\sin \left(1\right)-\cos \left(1\right)\right)\sin \left(x\right)+3\left(\sin \left(1\right)+\cos \left(1\right)\right)\cos \left(x\right)\right)$

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

For more information on Maple 2021 changes, see Updates in Maple 2021.

The Student[ODEs][Solve][HighOrder] 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.