The GenerateSimilarODE command takes an ordinary differential equation (ODE) eqn with 1 dependent and 1 independent variable and returns a similar ODE in the same variables.

Linear ordinary differential equations with constant coefficients that have order higher than 1 return a linear ordinary differential equations with constant coefficients that have similar roots to the characteristic polynomial of the ODE. Each real root in eqn will have a corresponding real root in the output ODE, each repeated root in eqn will correspond to a repeated root in the output ODE. A pair of complex conjugate roots in eqn will correspond to a pair of complex conjugate roots in the output ODE.

Linear ordinary differential equations with constant coefficients that have order higher than 1 and a forcing function that contains functions that are linearly dependent to the solution of the homogeneous ODE produce an ODE with the similar roots described above and a forcing function that has functions that are linearly dependent to the solutions of the homogeneous output ODE.   

Bessel differential equations or differential equations that can be converted into Bessel differential equations return Bessel differential equations or differential equations that can be converted into Bessel differential equations.

Differential equations that when solved produce terminating Legendre polynomials return differential equations that when solved produce terminating Legendre polynomials.

Differential equations that when solved produce terminating Laguerre polynomials return differential equations that when solved produce terminating Laguerre polynomials.

Chebyshev differential equations produce Chebyshev differential equations.

$\mathrm{with}\left(\mathrm{RandomTools}\right)\:$

$\mathrm{ODE1}≔\mathrm{\%diff}\left(y\left(x\right)\,x\right)y\left(x\right)+\sin \left(x\right)=\exp \left(x\right)y\left(x\right)$

$\mathrm{ODE1}≔\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)+\sin \left(x\right)={\ⅇ}^{x}y\left(x\right)$

$\mathrm{GenerateSimilarODE}\left(\mathrm{ODE1}\right)$

$4\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)-9\cos \left(x\right)=-7{\ⅇ}^{-6x}y\left(x\right)$

$\mathrm{ODE2}≔\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\mathrm{\%diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ODE2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{\ⅆ}{\ⅆx}y\left(x\right)-6y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE2}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{2x}+\mathrm{c\_\_2}{\ⅇ}^{-3x}$

$\mathrm{newODE2}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE2}\right)$

$\mathrm{newODE2}≔19\frac{\ⅆ}{\ⅆx}y\left(x\right)-\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-90y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE2}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{9x}+\mathrm{c\_\_2}{\ⅇ}^{10x}$

$\mathrm{ODE3}≔\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-6\mathrm{\%diff}\left(y\left(x\right)\,x\right)+9y\left(x\right)=0$

$\mathrm{ODE3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-6\frac{\ⅆ}{\ⅆx}y\left(x\right)+9y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE3}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{3x}+\mathrm{c\_\_2}{\ⅇ}^{3x}x$

$\mathrm{newODE3}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE3}\right)$

$\mathrm{newODE3}≔-16\frac{\ⅆ}{\ⅆx}y\left(x\right)+\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+64y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE3}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{8x}+\mathrm{c\_\_2}{\ⅇ}^{8x}x$

$\mathrm{ODE4}≔\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-2\mathrm{\%diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=0$

$\mathrm{ODE4}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-2\frac{\ⅆ}{\ⅆx}y\left(x\right)+2y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE4}\right)$

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

$\mathrm{newODE4}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE4}\right)$

$\mathrm{newODE4}≔-20\frac{\ⅆ}{\ⅆx}y\left(x\right)+\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+101y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE4}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{10x}\sin \left(x\right)+\mathrm{c\_\_2}{\ⅇ}^{10x}\cos \left(x\right)$

$\mathrm{ODE5}≔\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\mathrm{\%diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=x\exp \left(2x\right)$

$\mathrm{ODE5}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{\ⅆ}{\ⅆx}y\left(x\right)-6y\left(x\right)=x{\ⅇ}^{2x}$

$\mathrm{dsolve}\left(\mathrm{ODE5}\right)$

$y\left(x\right)={\ⅇ}^{2x}\mathrm{c\_\_2}+{\ⅇ}^{-3x}\mathrm{c\_\_1}+\frac{{\ⅇ}^{2x}x\left(5x-2\right)}{50}$

$\mathrm{newODE5}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE5}\right)$

$\mathrm{newODE5}≔-18\frac{\ⅆ}{\ⅆx}y\left(x\right)-\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-80y\left(x\right)=-10x{\ⅇ}^{-8x}$

$\mathrm{dsolve}\left(\mathrm{newODE5}\right)$

$y\left(x\right)={\ⅇ}^{-8x}\mathrm{c\_\_2}+{\ⅇ}^{-10x}\mathrm{c\_\_1}+\frac{5x\left(x-1\right){\ⅇ}^{-8x}}{2}$

$\mathrm{ODE6}≔x^2\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+x\mathrm{\%diff}\left(y\left(x\right)\,x\right)+x^{2}y\left(x\right)=0$

$\mathrm{ODE6}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+x^{2}y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE6}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{BesselJ}\left(0\,x\right)+\mathrm{c\_\_2}\mathrm{BesselY}\left(0\,x\right)$

$\mathrm{newODE6}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE6}\right)$

$\mathrm{newODE6}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-36\right)y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE6}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{BesselJ}\left(6\,x\right)+\mathrm{c\_\_2}\mathrm{BesselY}\left(6\,x\right)$

$\mathrm{ODE7}≔x^2\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+x\mathrm{\%diff}\left(y\left(x\right)\,x\right)+\left(x^2-9\right)y\left(x\right)=0$

$\mathrm{ODE7}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-9\right)y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE7}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{BesselJ}\left(3\,x\right)+\mathrm{c\_\_2}\mathrm{BesselY}\left(3\,x\right)$

$\mathrm{newODE7}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE7}\right)$

$\mathrm{newODE7}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-49\right)y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE7}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\mathrm{BesselJ}\left(7\,x\right)+\mathrm{c\_\_2}\mathrm{BesselY}\left(7\,x\right)$

$\mathrm{ODE8}≔x^2\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+2x\mathrm{\%diff}\left(y\left(x\right)\,x\right)+x^{2}y\left(x\right)=0$

$\mathrm{ODE8}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+x^{2}y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE8}\right)$

$y\left(x\right)=\frac{\mathrm{c\_\_1}\sin \left(x\right)}{x}+\frac{\mathrm{c\_\_2}\cos \left(x\right)}{x}$

$\mathrm{newODE8}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE8}\right)$

$\mathrm{newODE8}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+3x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-4\right)y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE8}\right)$

$y\left(x\right)=\frac{\mathrm{c\_\_1}\mathrm{BesselJ}\left(\sqrt{5}\,x\right)}{x}+\frac{\mathrm{c\_\_2}\mathrm{BesselY}\left(\sqrt{5}\,x\right)}{x}$

$\mathrm{ODE9}≔2x^2\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+x\mathrm{\%diff}\left(y\left(x\right)\,x\right)+x^{2}y\left(x\right)=0$

$\mathrm{ODE9}≔2x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+x^{2}y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE9}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{BesselJ}\left(\frac{1}{4}\,\frac{\sqrt{2}x}{2}\right)+\mathrm{c\_\_2}{x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{BesselY}\left(\frac{1}{4}\,\frac{\sqrt{2}x}{2}\right)$

$\mathrm{newODE9}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE9}\right)$

$\mathrm{newODE9}≔5x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-4\right)y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE9}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{x}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\mathrm{BesselJ}\left(\frac{2\sqrt{6}}{5}\,\frac{\sqrt{5}x}{5}\right)+\mathrm{c\_\_2}{x}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\mathrm{BesselY}\left(\frac{2\sqrt{6}}{5}\,\frac{\sqrt{5}x}{5}\right)$

$\mathrm{ODE10}≔x\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\left(1-x\right)\mathrm{\%diff}\left(y\left(x\right)\,x\right)+y\left(x\right)=0$

$\mathrm{ODE10}≔x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(1-x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE10}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(x-1\right)+\mathrm{c\_\_2}\left(\left(x-1\right){\mathrm{Ei}}_1\left(-x\right)+{\ⅇ}^x\right)$

$\mathrm{newODE10}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE10}\right)$

$\mathrm{newODE10}≔x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(1-x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+2y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE10}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(x^2-4x+2\right)+\mathrm{c\_\_2}\left(\frac{\left(x^2-4x+2\right){\mathrm{Ei}}_1\left(-x\right)}{4}+\frac{\left(x-3\right){\ⅇ}^x}{4}\right)$

$\mathrm{ODE11}≔x\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\left(1-x\right)\mathrm{\%diff}\left(y\left(x\right)\,x\right)+5y\left(x\right)=0$

$\mathrm{ODE11}≔x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(1-x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+5y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE11}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(x^5-25x^4+200x^3-600x^2+600x-120\right)+\mathrm{c\_\_2}\left(\frac{\left(x^5-25x^4+200x^3-600x^2+600x-120\right){\mathrm{Ei}}_1\left(-x\right)}{600}+\frac{{\ⅇ}^x\left(x^4-24x^3+177x^2-444x+274\right)}{600}\right)$

$\mathrm{newODE11}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE11}\right)$

$\mathrm{newODE11}≔x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(1-x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE11}\right)$

$y\left(x\right)=\mathrm{c\_\_1}+{\mathrm{Ei}}_1\left(-x\right)\mathrm{c\_\_2}$

$\mathrm{ODE12}≔\left(1-x^2\right)\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-2x\mathrm{\%diff}\left(y\left(x\right)\,x\right)+6y\left(x\right)=0$

$\mathrm{ODE12}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+6y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE12}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(-3x^2+1\right)+\mathrm{c\_\_2}\left(\frac{\left(3x^2-1\right)\ln \left(x-1\right)}{2}+\frac{\left(-3x^2+1\right)\ln \left(x+1\right)}{2}+3x\right)$

$\mathrm{newODE12}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE12}\right)$

$\mathrm{newODE12}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-\frac{y\left(x\right)}{-x^2+1}=0$

$\mathrm{dsolve}\left(\mathrm{newODE12}\right)$

$y\left(x\right)=\frac{\mathrm{c\_\_1}x}{\sqrt{-x^2+1}}+\frac{\mathrm{c\_\_2}}{\sqrt{-x^2+1}}$

$\mathrm{ODE13}≔\left(1-x^2\right)\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-2x\mathrm{\%diff}\left(y\left(x\right)\,x\right)+12y\left(x\right)=0$

$\mathrm{ODE13}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+12y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE13}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(-\frac{5}{3}{x}^3+x\right)+\mathrm{c\_\_2}\left(-\frac{1}{9}+\frac{\left(5x^3-3x\right)\ln \left(x-1\right)}{24}+\frac{\left(-5x^3+3x\right)\ln \left(x+1\right)}{24}+\frac{5x^2}{12}\right)$

$\mathrm{newODE13}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE13}\right)$

$\mathrm{newODE13}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+20y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE13}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(\frac{35}{3}{x}^4-10x^2+1\right)+\mathrm{c\_\_2}\left(\frac{\left(35x^4-30x^2+3\right)\ln \left(x-1\right)}{6}+\frac{\left(-35x^4+30x^2-3\right)\ln \left(x+1\right)}{6}+\frac{35x^3}{3}-\frac{55x}{9}\right)$

$\mathrm{ODE14}≔\left(1-x^2\right)\mathrm{\%diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-x\mathrm{\%diff}\left(y\left(x\right)\,x\right)+25y\left(x\right)=0$

$\mathrm{ODE14}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+25y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{ODE14}\right)$

$y\left(x\right)=\frac{\mathrm{c\_\_1}}{{\left(x+\sqrt{x^2-1}\right)}^5}+\mathrm{c\_\_2}{\left(x+\sqrt{x^2-1}\right)}^5$

$\mathrm{newODE14}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE14}\right)$

$\mathrm{newODE14}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+36y\left(x\right)=0$

$\mathrm{dsolve}\left(\mathrm{newODE14}\right)$

$y\left(x\right)=\frac{\mathrm{c\_\_1}}{{\left(x+\sqrt{x^2-1}\right)}^6}+\mathrm{c\_\_2}{\left(x+\sqrt{x^2-1}\right)}^6$

The RandomTools[GenerateSimilarODE] command was introduced in Maple 2021.

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