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

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

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

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

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

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

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

$y\left(x\right)=3\mathrm{\_C2}{x}^3-\frac{\mathrm{\_C1}}{x}-\frac{2\mathrm{\_C3}}{x^2}$

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

$y\left(x\right)=\frac{7x^3}{20}+\frac{13}{4x}-\frac{8}{5x^2}$

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

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

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

$y\left(x\right)=-{\ⅇ}^{-x}\left(\left(\mathrm{\_C2}+\mathrm{\_C3}\right)\sin \left(x\right)+\left(\mathrm{\_C2}-\mathrm{\_C3}\right)\cos \left(x\right)+\mathrm{\_C1}\right)$

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

$y\left(x\right)=-{\ⅇ}^{-x}\left(\left(\frac{3\sin \left(1\right)}{{\ⅇ}^{\mathrm{-1}}\left({\cos \left(1\right)}^2+{\sin \left(1\right)}^2\right)}-\frac{3\cos \left(1\right)}{{\ⅇ}^{\mathrm{-1}}\left({\cos \left(1\right)}^2+{\sin \left(1\right)}^2\right)}\right)\sin \left(x\right)+\left(\frac{3\sin \left(1\right)}{{\ⅇ}^{\mathrm{-1}}\left({\cos \left(1\right)}^2+{\sin \left(1\right)}^2\right)}+\frac{3\cos \left(1\right)}{{\ⅇ}^{\mathrm{-1}}\left({\cos \left(1\right)}^2+{\sin \left(1\right)}^2\right)}\right)\cos \left(x\right)-\frac{5}{{\ⅇ}^{\mathrm{-1}}}\right)$

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

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