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RandomTools

 GenerateSimilarODE
 create a random differential equation similar to the one given

 Calling Sequence GenerateSimilarODE( eqn )

Parameters

 eqn - differential equation with one dependent and one independent variable

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{RandomTools}\right):$
 > $\mathrm{ODE1}≔\mathrm{%diff}\left(y\left(x\right),x\right)y\left(x\right)+\mathrm{sin}\left(x\right)=\mathrm{exp}\left(x\right)y\left(x\right)$
 ${\mathrm{ODE1}}{≔}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){}{y}{}\left({x}\right){+}{\mathrm{sin}}{}\left({x}\right){=}{{ⅇ}}^{{x}}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{GenerateSimilarODE}\left(\mathrm{ODE1}\right)$
 ${4}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){}{y}{}\left({x}\right){-}{9}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%cos}}{}\left({x}\right)\right]\right){=}{-}{7}{}{{ⅇ}}^{{-}{6}{}{x}}{}{y}{}\left({x}\right)$ (2)

2nd order linear ODE with constant coefficients with a characteristic polynomial that has real roots.

 > $\mathrm{ODE2}≔\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+\mathrm{%diff}\left(y\left(x\right),x\right)-6y\left(x\right)=0$
 ${\mathrm{ODE2}}{≔}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){+}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (3)
 > $\mathrm{dsolve}\left(\mathrm{ODE2}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{2}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{3}{}{x}}$ (4)
 > $\mathrm{newODE2}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE2}\right)$
 ${\mathrm{newODE2}}{≔}{19}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){-}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){-}{90}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{dsolve}\left(\mathrm{newODE2}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{9}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{10}{}{x}}$ (6)

2nd order linear ODE with constant coefficients with a repeated root.

 > $\mathrm{ODE3}≔\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)-6\mathrm{%diff}\left(y\left(x\right),x\right)+9y\left(x\right)=0$
 ${\mathrm{ODE3}}{≔}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){-}{6}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){+}{9}{}{y}{}\left({x}\right){=}{0}$ (7)
 > $\mathrm{dsolve}\left(\mathrm{ODE3}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{3}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{3}{}{x}}{}{x}$ (8)
 > $\mathrm{newODE3}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE3}\right)$
 ${\mathrm{newODE3}}{≔}{-}{16}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){+}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){+}{64}{}{y}{}\left({x}\right){=}{0}$ (9)
 > $\mathrm{dsolve}\left(\mathrm{newODE3}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{8}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{8}{}{x}}{}{x}$ (10)

2nd order linear ODE with a pair of complex conjugate roots.

 > $\mathrm{ODE4}≔\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)-2\mathrm{%diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{ODE4}}{≔}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){-}{2}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (11)
 > $\mathrm{dsolve}\left(\mathrm{ODE4}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)$ (12)
 > $\mathrm{newODE4}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE4}\right)$
 ${\mathrm{newODE4}}{≔}{-}{20}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){+}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){+}{101}{}{y}{}\left({x}\right){=}{0}$ (13)
 > $\mathrm{dsolve}\left(\mathrm{newODE4}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{10}{}{x}}{}{\mathrm{sin}}{}\left({x}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{{10}{}{x}}{}{\mathrm{cos}}{}\left({x}\right)$ (14)

2nd order linear ODE with forcing function that contains a function that is linearly dependent to a solution to the homogeneous ODE.

 > $\mathrm{ODE5}≔\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+\mathrm{%diff}\left(y\left(x\right),x\right)-6y\left(x\right)=x\mathrm{exp}\left(2x\right)$
 ${\mathrm{ODE5}}{≔}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){+}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){-}{6}{}{y}{}\left({x}\right){=}{x}{}{{ⅇ}}^{{2}{}{x}}$ (15)
 > $\mathrm{dsolve}\left(\mathrm{ODE5}\right)$
 ${y}{}\left({x}\right){=}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{_C2}}{+}{{ⅇ}}^{{-}{3}{}{x}}{}{\mathrm{_C1}}{+}\frac{{x}{}{{ⅇ}}^{{2}{}{x}}{}\left({5}{}{x}{-}{2}\right)}{{50}}$ (16)
 > $\mathrm{newODE5}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE5}\right)$
 ${\mathrm{newODE5}}{≔}{-}{18}{}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right){-}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right){-}{80}{}{y}{}\left({x}\right){=}{-}{10}{}{x}{}{{ⅇ}}^{{-}{8}{}{x}}$ (17)
 > $\mathrm{dsolve}\left(\mathrm{newODE5}\right)$
 ${y}{}\left({x}\right){=}{{ⅇ}}^{{-}{8}{}{x}}{}{\mathrm{_C2}}{+}{{ⅇ}}^{{-}{10}{}{x}}{}{\mathrm{_C1}}{+}\frac{{5}{}{x}{}\left({x}{-}{1}\right){}{{ⅇ}}^{{-}{8}{}{x}}}{{2}}$ (18)

Bessel differential equation.

 > $\mathrm{ODE6}≔{x}^{2}\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+x\mathrm{%diff}\left(y\left(x\right),x\right)+{x}^{2}y\left(x\right)=0$
 ${\mathrm{ODE6}}{≔}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{{x}}^{{2}}{}{y}{}\left({x}\right){=}{0}$ (19)
 > $\mathrm{dsolve}\left(\mathrm{ODE6}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left({0}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left({0}{,}{x}\right)$ (20)
 > $\mathrm{newODE6}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE6}\right)$
 ${\mathrm{newODE6}}{≔}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}\left({{x}}^{{2}}{-}{36}\right){}{y}{}\left({x}\right){=}{0}$ (21)
 > $\mathrm{dsolve}\left(\mathrm{newODE6}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left({6}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left({6}{,}{x}\right)$ (22)
 > $\mathrm{ODE7}≔{x}^{2}\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+x\mathrm{%diff}\left(y\left(x\right),x\right)+\left({x}^{2}-9\right)y\left(x\right)=0$
 ${\mathrm{ODE7}}{≔}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}\left({{x}}^{{2}}{-}{9}\right){}{y}{}\left({x}\right){=}{0}$ (23)
 > $\mathrm{dsolve}\left(\mathrm{ODE7}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left({3}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left({3}{,}{x}\right)$ (24)
 > $\mathrm{newODE7}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE7}\right)$
 ${\mathrm{newODE7}}{≔}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}\left({{x}}^{{2}}{-}{49}\right){}{y}{}\left({x}\right){=}{0}$ (25)
 > $\mathrm{dsolve}\left(\mathrm{newODE7}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left({7}{,}{x}\right){+}{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left({7}{,}{x}\right)$ (26)

ODEs that can be converted to a Bessel differential equation.

 > $\mathrm{ODE8}≔{x}^{2}\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+2x\mathrm{%diff}\left(y\left(x\right),x\right)+{x}^{2}y\left(x\right)=0$
 ${\mathrm{ODE8}}{≔}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{2}{}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{{x}}^{{2}}{}{y}{}\left({x}\right){=}{0}$ (27)
 > $\mathrm{dsolve}\left(\mathrm{ODE8}\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}{\mathrm{sin}}{}\left({x}\right)}{{x}}{+}\frac{{\mathrm{_C2}}{}{\mathrm{cos}}{}\left({x}\right)}{{x}}$ (28)
 > $\mathrm{newODE8}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE8}\right)$
 ${\mathrm{newODE8}}{≔}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{3}{}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}\left({{x}}^{{2}}{-}{4}\right){}{y}{}\left({x}\right){=}{0}$ (29)
 > $\mathrm{dsolve}\left(\mathrm{newODE8}\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}{\mathrm{BesselJ}}{}\left(\sqrt{{5}}{,}{x}\right)}{{x}}{+}\frac{{\mathrm{_C2}}{}{\mathrm{BesselY}}{}\left(\sqrt{{5}}{,}{x}\right)}{{x}}$ (30)
 > $\mathrm{ODE9}≔2{x}^{2}\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+x\mathrm{%diff}\left(y\left(x\right),x\right)+{x}^{2}y\left(x\right)=0$
 ${\mathrm{ODE9}}{≔}{2}{}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{{x}}^{{2}}{}{y}{}\left({x}\right){=}{0}$ (31)
 > $\mathrm{dsolve}\left(\mathrm{ODE9}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{x}}^{{1}}{{4}}}{}{\mathrm{BesselJ}}{}\left(\frac{{1}}{{4}}{,}\frac{\sqrt{{2}}{}{x}}{{2}}\right){+}{\mathrm{_C2}}{}{{x}}^{{1}}{{4}}}{}{\mathrm{BesselY}}{}\left(\frac{{1}}{{4}}{,}\frac{\sqrt{{2}}{}{x}}{{2}}\right)$ (32)
 > $\mathrm{newODE9}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE9}\right)$
 ${\mathrm{newODE9}}{≔}{5}{}{{x}}^{{2}}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}\left({{x}}^{{2}}{-}{4}\right){}{y}{}\left({x}\right){=}{0}$ (33)
 > $\mathrm{dsolve}\left(\mathrm{newODE9}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{x}}^{{2}}{{5}}}{}{\mathrm{BesselJ}}{}\left(\frac{{2}{}\sqrt{{6}}}{{5}}{,}\frac{\sqrt{{5}}{}{x}}{{5}}\right){+}{\mathrm{_C2}}{}{{x}}^{{2}}{{5}}}{}{\mathrm{BesselY}}{}\left(\frac{{2}{}\sqrt{{6}}}{{5}}{,}\frac{\sqrt{{5}}{}{x}}{{5}}\right)$ (34)

Terminating Laguerre polynomials.

 > $\mathrm{ODE10}≔x\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+\left(1-x\right)\mathrm{%diff}\left(y\left(x\right),x\right)+y\left(x\right)=0$
 ${\mathrm{ODE10}}{≔}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}\left({-}{x}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{y}{}\left({x}\right){=}{0}$ (35)
 > $\mathrm{dsolve}\left(\mathrm{ODE10}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({x}{-}{1}\right){+}{\mathrm{_C2}}{}\left(\left({x}{-}{1}\right){}{{\mathrm{Ei}}}_{{1}}{}\left({-}{x}\right){+}{{ⅇ}}^{{x}}\right)$ (36)
 > $\mathrm{newODE10}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE10}\right)$
 ${\mathrm{newODE10}}{≔}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}\left({-}{x}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (37)
 > $\mathrm{dsolve}\left(\mathrm{newODE10}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({{x}}^{{2}}{-}{4}{}{x}{+}{2}\right){+}{\mathrm{_C2}}{}\left(\frac{\left({{x}}^{{2}}{-}{4}{}{x}{+}{2}\right){}{{\mathrm{Ei}}}_{{1}}{}\left({-}{x}\right)}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\left({x}{-}{3}\right)}{{4}}\right)$ (38)
 > $\mathrm{ODE11}≔x\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+\left(1-x\right)\mathrm{%diff}\left(y\left(x\right),x\right)+5y\left(x\right)=0$
 ${\mathrm{ODE11}}{≔}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}\left({-}{x}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{5}{}{y}{}\left({x}\right){=}{0}$ (39)
 > $\mathrm{dsolve}\left(\mathrm{ODE11}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({{x}}^{{5}}{-}{25}{}{{x}}^{{4}}{+}{200}{}{{x}}^{{3}}{-}{600}{}{{x}}^{{2}}{+}{600}{}{x}{-}{120}\right){+}{\mathrm{_C2}}{}\left(\frac{\left({{x}}^{{5}}{-}{25}{}{{x}}^{{4}}{+}{200}{}{{x}}^{{3}}{-}{600}{}{{x}}^{{2}}{+}{600}{}{x}{-}{120}\right){}{{\mathrm{Ei}}}_{{1}}{}\left({-}{x}\right)}{{14400}}{+}\frac{{{ⅇ}}^{{x}}{}\left({{x}}^{{4}}{-}{24}{}{{x}}^{{3}}{+}{177}{}{{x}}^{{2}}{-}{444}{}{x}{+}{274}\right)}{{14400}}\right)$ (40)
 > $\mathrm{newODE11}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE11}\right)$
 ${\mathrm{newODE11}}{≔}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){+}\left({-}{x}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){=}{0}$ (41)
 > $\mathrm{dsolve}\left(\mathrm{newODE11}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{+}{{\mathrm{Ei}}}_{{1}}{}\left({-}{x}\right){}{\mathrm{_C2}}$ (42)

Terminating Legendre polynomials.

 > $\mathrm{ODE12}≔\left(1-{x}^{2}\right)\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)-2x\mathrm{%diff}\left(y\left(x\right),x\right)+6y\left(x\right)=0$
 ${\mathrm{ODE12}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){-}{2}{}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{6}{}{y}{}\left({x}\right){=}{0}$ (43)
 > $\mathrm{dsolve}\left(\mathrm{ODE12}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({-}{3}{}{{x}}^{{2}}{+}{1}\right){+}{\mathrm{_C2}}{}\left(\left(\frac{{3}{}{{x}}^{{2}}}{{8}}{-}\frac{{1}}{{8}}\right){}{\mathrm{ln}}{}\left({x}{-}{1}\right){+}\left({-}\frac{{3}{}{{x}}^{{2}}}{{8}}{+}\frac{{1}}{{8}}\right){}{\mathrm{ln}}{}\left({x}{+}{1}\right){+}\frac{{3}{}{x}}{{4}}\right)$ (44)
 > $\mathrm{newODE12}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE12}\right)$
 ${\mathrm{newODE12}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){-}{2}{}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){-}\frac{{y}{}\left({x}\right)}{{-}{{x}}^{{2}}{+}{1}}{=}{0}$ (45)
 > $\mathrm{dsolve}\left(\mathrm{newODE12}\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}{x}}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{+}\frac{{\mathrm{_C2}}}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}$ (46)
 > $\mathrm{ODE13}≔\left(1-{x}^{2}\right)\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)-2x\mathrm{%diff}\left(y\left(x\right),x\right)+12y\left(x\right)=0$
 ${\mathrm{ODE13}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){-}{2}{}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{12}{}{y}{}\left({x}\right){=}{0}$ (47)
 > $\mathrm{dsolve}\left(\mathrm{ODE13}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({-}\frac{{5}}{{3}}{}{{x}}^{{3}}{+}{x}\right){+}{\mathrm{_C2}}{}\left(\frac{\left({15}{}{{x}}^{{3}}{-}{9}{}{x}\right){}{\mathrm{ln}}{}\left({x}{-}{1}\right)}{{72}}{+}\frac{\left({-}{15}{}{{x}}^{{3}}{+}{9}{}{x}\right){}{\mathrm{ln}}{}\left({x}{+}{1}\right)}{{72}}{+}\frac{{5}{}{{x}}^{{2}}}{{12}}{-}\frac{{1}}{{9}}\right)$ (48)
 > $\mathrm{newODE13}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE13}\right)$
 ${\mathrm{newODE13}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){-}{2}{}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{20}{}{y}{}\left({x}\right){=}{0}$ (49)
 > $\mathrm{dsolve}\left(\mathrm{newODE13}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left(\frac{{35}}{{3}}{}{{x}}^{{4}}{-}{10}{}{{x}}^{{2}}{+}{1}\right){+}{\mathrm{_C2}}{}\left(\frac{\left({105}{}{{x}}^{{4}}{-}{90}{}{{x}}^{{2}}{+}{9}\right){}{\mathrm{ln}}{}\left({x}{-}{1}\right)}{{1152}}{+}\frac{\left({-}{105}{}{{x}}^{{4}}{+}{90}{}{{x}}^{{2}}{-}{9}\right){}{\mathrm{ln}}{}\left({x}{+}{1}\right)}{{1152}}{+}\frac{{35}{}{{x}}^{{3}}}{{192}}{-}\frac{{55}{}{x}}{{576}}\right)$ (50)

Chebyshev differential equation.

 > $\mathrm{ODE14}≔\left(1-{x}^{2}\right)\mathrm{%diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)-x\mathrm{%diff}\left(y\left(x\right),x\right)+25y\left(x\right)=0$
 ${\mathrm{ODE14}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){-}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{25}{}{y}{}\left({x}\right){=}{0}$ (51)
 > $\mathrm{dsolve}\left(\mathrm{ODE14}\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}}{{\left({x}{+}\sqrt{{{x}}^{{2}}{-}{1}}\right)}^{{5}}}{+}{\mathrm{_C2}}{}{\left({x}{+}\sqrt{{{x}}^{{2}}{-}{1}}\right)}^{{5}}$ (52)
 > $\mathrm{newODE14}≔\mathrm{GenerateSimilarODE}\left(\mathrm{ODE14}\right)$
 ${\mathrm{newODE14}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}{,}{x}\right)\right]\right)\right){-}{x}{}\left({\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%diff}}{}\left({y}{}\left({x}\right){,}{x}\right)\right]\right)\right){+}{36}{}{y}{}\left({x}\right){=}{0}$ (53)
 > $\mathrm{dsolve}\left(\mathrm{newODE14}\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}}{{\left({x}{+}\sqrt{{{x}}^{{2}}{-}{1}}\right)}^{{6}}}{+}{\mathrm{_C2}}{}{\left({x}{+}\sqrt{{{x}}^{{2}}{-}{1}}\right)}^{{6}}$ (54)

Compatibility

 • The RandomTools[GenerateSimilarODE] command was introduced in Maple 2021.
 • For more information on Maple 2021 changes, see Updates in Maple 2021.

 See Also