The SecondOrder(ODE, y(x)) command finds the solution of a second order ODE.

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

$\mathrm{ode1}≔2x\mathrm{diff}\left(y\left(x\right)\,x\right)-9x^2+\left(2\mathrm{diff}\left(y\left(x\right)\,x\right)+x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=0$

$\mathrm{ode1}≔2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-9x^2+\left(2\frac{\ⅆ}{\ⅆx}y\left(x\right)+x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)=0$

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

$\left\{y\left(x\right)=\int \left(-\frac{x^2}{2}-\frac{1}{2}-\frac{\sqrt{x^4+12x^3+2x^2+4\mathrm{\_C1}+1}}{2}\right)\ⅆx+\mathrm{\_C2}\,y\left(x\right)=\int \left(-\frac{x^2}{2}-\frac{1}{2}+\frac{\sqrt{x^4+12x^3+2x^2+4\mathrm{\_C1}+1}}{2}\right)\ⅆx+\mathrm{\_C2}\right\}$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)-x\exp \left(x\right)=0$

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

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

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

$\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\frac{5{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2}{y\left(x\right)}=0$

$\mathrm{ode3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{5{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{y\left(x\right)}=0$

$\mathrm{SecondOrder}\left(\mathrm{ode3}\,y\left(x\right)\right)$

$\left\{y\left(x\right)={\left(6{\ⅇ}^{\mathrm{\_C1}}x+6\mathrm{\_C2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\,y\left(x\right)=-{\left(6{\ⅇ}^{\mathrm{\_C1}}x+6\mathrm{\_C2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right\}$

$\mathrm{ode4}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ode4}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-\frac{\ⅆ}{\ⅆx}y\left(x\right)-6y\left(x\right)=0$

$\mathrm{SecondOrder}\left(\mathrm{ode4}\,y\left(x\right)\right)$

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

$\mathrm{ode5}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)=x^2+6y\left(x\right)$

$\mathrm{ode5}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-\frac{\ⅆ}{\ⅆx}y\left(x\right)=x^2+6y\left(x\right)$

$\mathrm{SecondOrder}\left(\mathrm{ode5}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-2x}+\mathrm{\_C2}{\ⅇ}^{3x}-\frac{x^2}{6}+\frac{x}{18}-\frac{7}{108}$

$\mathrm{ode6}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4y\left(x\right)=-4\mathrm{diff}\left(y\left(x\right)\,x\right)$

$\mathrm{ode6}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4y\left(x\right)=-4\frac{\ⅆ}{\ⅆx}y\left(x\right)$

$\mathrm{SecondOrder}\left(\mathrm{ode6}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-2x}+\mathrm{\_C2}x{\ⅇ}^{-2x}$

$\mathrm{ode7}≔5\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+20y\left(x\right)+15\sin \left(x\right)=-20\mathrm{diff}\left(y\left(x\right)\,x\right)$

$\mathrm{ode7}≔5\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+20y\left(x\right)+15\sin \left(x\right)=-20\frac{\ⅆ}{\ⅆx}y\left(x\right)$

$\mathrm{SecondOrder}\left(\mathrm{ode7}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-2x}+\mathrm{\_C2}x{\ⅇ}^{-2x}+\frac{12\cos \left(x\right)}{25}-\frac{9\sin \left(x\right)}{25}$

$\mathrm{ode8}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+2y\left(x\right)+2\mathrm{diff}\left(y\left(x\right)\,x\right)=0$

$\mathrm{ode8}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+2y\left(x\right)+2\frac{\ⅆ}{\ⅆx}y\left(x\right)=0$

$\mathrm{SecondOrder}\left(\mathrm{ode8}\,y\left(x\right)\right)$

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

$\mathrm{ode9}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+2y\left(x\right)-2\mathrm{diff}\left(y\left(x\right)\,x\right)=\exp \left(x\right)$

$\mathrm{ode9}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+2y\left(x\right)-2\frac{\ⅆ}{\ⅆx}y\left(x\right)={\ⅇ}^x$

$\mathrm{SecondOrder}\left(\mathrm{ode9}\,y\left(x\right)\right)$

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

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

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