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.

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

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

$\mathrm{ODE1}≔\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)y\left(x\right)+\sin \left(x\right)={\ⅇ}^{x}y\left(x\right)$

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

$4\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)y\left(x\right)-9\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%cos}\left(x\right)\right]\right)=-7{\ⅇ}^{-6x}y\left(x\right)$

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

$\mathrm{ODE2}≔\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)+\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)-6y\left(x\right)=0$

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

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

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

$\mathrm{newODE2}≔19\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)-\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)-90y\left(x\right)=0$

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

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

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

$\mathrm{ODE3}≔\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)-6\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)+9y\left(x\right)=0$

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

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

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

$\mathrm{newODE3}≔-16\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)+\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)+64y\left(x\right)=0$

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

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

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

$\mathrm{ODE4}≔\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)-2\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)+2y\left(x\right)=0$

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

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

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

$\mathrm{newODE4}≔-20\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)+\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)+101y\left(x\right)=0$

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

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

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

$\mathrm{ODE5}≔\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)+\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)-6y\left(x\right)=x{\ⅇ}^{2x}$

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

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

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

$\mathrm{newODE5}≔-18\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)-\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)-80y\left(x\right)=-10x{\ⅇ}^{-8x}$

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

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

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

$\mathrm{ODE6}≔x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+x^{2}y\left(x\right)=0$

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

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

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

$\mathrm{newODE6}≔x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+\left(x^2-36\right)y\left(x\right)=0$

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

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

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

$\mathrm{ODE7}≔x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+\left(x^2-9\right)y\left(x\right)=0$

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

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

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

$\mathrm{newODE7}≔x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+\left(x^2-49\right)y\left(x\right)=0$

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

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

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

$\mathrm{ODE8}≔x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+2x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+x^{2}y\left(x\right)=0$

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

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

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

$\mathrm{newODE8}≔x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+3x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+\left(x^2-4\right)y\left(x\right)=0$

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

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

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

$\mathrm{ODE9}≔2x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+x^{2}y\left(x\right)=0$

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

$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)$

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

$\mathrm{newODE9}≔5x^2\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+\left(x^2-4\right)y\left(x\right)=0$

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

$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)$

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

$\mathrm{ODE10}≔x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+\left(1-x\right)\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+y\left(x\right)=0$

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

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

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

$\mathrm{newODE10}≔x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+\left(1-x\right)\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+2y\left(x\right)=0$

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

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

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

$\mathrm{ODE11}≔x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+\left(1-x\right)\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+5y\left(x\right)=0$

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

$y\left(x\right)=\mathrm{c\_\_1}\left(x^5-25x^4+200x^3-600x^2+600x-120\right)+\mathrm{c\_\_2}\left(\frac{\left(x^5-25x^4+200x^3-600x^2+600x-120\right){\mathrm{Ei}}_1\left(-x\right)}{14400}+\frac{{\ⅇ}^x\left(x^4-24x^3+177x^2-444x+274\right)}{14400}\right)$

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

$\mathrm{newODE11}≔x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)+\left(1-x\right)\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)=0$

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

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

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

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

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

$y\left(x\right)=\mathrm{c\_\_1}\left(-3x^2+1\right)+\mathrm{c\_\_2}\left(\left(\frac{3x^2}{8}-\frac{1}{8}\right)\ln \left(x-1\right)+\left(-\frac{3x^2}{8}+\frac{1}{8}\right)\ln \left(x+1\right)+\frac{3x}{4}\right)$

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

$\mathrm{newODE12}≔\left(-x^2+1\right)\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)-2x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)-\frac{y\left(x\right)}{-x^2+1}=0$

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

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

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

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

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

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

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

$\mathrm{newODE13}≔\left(-x^2+1\right)\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)-2x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+20y\left(x\right)=0$

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

$y\left(x\right)=\mathrm{c\_\_1}\left(\frac{35}{3}{x}^4-10x^2+1\right)+\mathrm{c\_\_2}\left(\frac{\left(35x^4-30x^2+3\right)\ln \left(x-1\right)}{384}+\frac{\left(-35x^4+30x^2-3\right)\ln \left(x+1\right)}{384}+\frac{35x^3}{192}-\frac{55x}{576}\right)$

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

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

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

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

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

$\mathrm{newODE14}≔\left(-x^2+1\right)\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\,x\right)\right]\right)\right)-x\left(\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%diff}\left(y\left(x\right)\,x\right)\right]\right)\right)+36y\left(x\right)=0$

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

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

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

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