RandomTools
GenerateSimilarODE
create a random differential equation similar to the one given
Calling Sequence
Parameters
Description
Examples
Compatibility
GenerateSimilarODE( eqn )
eqn
-
differential equation with one dependent and one independent variable
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.
with⁡RandomTools:
ODE1 ≔ %diff⁡y⁡x,x⁢y⁡x+sin⁡x=ⅇx⁢y⁡x
ODE1≔Typesetting:-_Hold⁡%diff⁡y⁡x,x⁢y⁡x+sin⁡x=ⅇx⁢y⁡x
GenerateSimilarODE⁡ODE1
4⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x⁢y⁡x−9⁢Typesetting:-_Hold⁡%cos⁡x=−7⁢ⅇ−6⁢x⁢y⁡x
2nd order linear ODE with constant coefficients with a characteristic polynomial that has real roots.
ODE2 ≔ %diff⁡y⁡x,x$2+%diff⁡y⁡x,x−6⁢y⁡x=0
ODE2≔Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+Typesetting:-_Hold⁡%diff⁡y⁡x,x−6⁢y⁡x=0
dsolve⁡ODE2
y⁡x=c__1⁢ⅇ2⁢x+c__2⁢ⅇ−3⁢x
newODE2 ≔ GenerateSimilarODE⁡ODE2
newODE2≔19⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x−Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−90⁢y⁡x=0
dsolve⁡newODE2
y⁡x=c__1⁢ⅇ9⁢x+c__2⁢ⅇ10⁢x
2nd order linear ODE with constant coefficients with a repeated root.
ODE3 ≔ %diff⁡y⁡x,x$2−6⁢%diff⁡y⁡x,x+9⁢y⁡x=0
ODE3≔Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−6⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+9⁢y⁡x=0
dsolve⁡ODE3
y⁡x=c__1⁢ⅇ3⁢x+c__2⁢ⅇ3⁢x⁢x
newODE3 ≔ GenerateSimilarODE⁡ODE3
newODE3≔−16⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+64⁢y⁡x=0
dsolve⁡newODE3
y⁡x=c__1⁢ⅇ8⁢x+c__2⁢ⅇ8⁢x⁢x
2nd order linear ODE with a pair of complex conjugate roots.
ODE4 ≔ %diff⁡y⁡x,x$2−2⁢%diff⁡y⁡x,x+2⁢y⁡x=0
ODE4≔Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+2⁢y⁡x=0
dsolve⁡ODE4
y⁡x=c__1⁢ⅇx⁢sin⁡x+c__2⁢ⅇx⁢cos⁡x
newODE4 ≔ GenerateSimilarODE⁡ODE4
newODE4≔−20⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+101⁢y⁡x=0
dsolve⁡newODE4
y⁡x=c__1⁢ⅇ10⁢x⁢sin⁡x+c__2⁢ⅇ10⁢x⁢cos⁡x
2nd order linear ODE with forcing function that contains a function that is linearly dependent to a solution to the homogeneous ODE.
ODE5 ≔ %diff⁡y⁡x,x$2+%diff⁡y⁡x,x−6⁢y⁡x=x⁢ⅇ2⁢x
ODE5≔Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+Typesetting:-_Hold⁡%diff⁡y⁡x,x−6⁢y⁡x=x⁢ⅇ2⁢x
dsolve⁡ODE5
y⁡x=ⅇ2⁢x⁢c__2+ⅇ−3⁢x⁢c__1+ⅇ2⁢x⁢x⁢5⁢x−250
newODE5 ≔ GenerateSimilarODE⁡ODE5
newODE5≔−18⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x−Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−80⁢y⁡x=−10⁢x⁢ⅇ−8⁢x
dsolve⁡newODE5
y⁡x=ⅇ−8⁢x⁢c__2+ⅇ−10⁢x⁢c__1+5⁢x⁢x−1⁢ⅇ−8⁢x2
Bessel differential equation.
ODE6 ≔ x2⁢%diff⁡y⁡x,x$2+x⁢%diff⁡y⁡x,x+x2⁢y⁡x=0
ODE6≔x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2⁢y⁡x=0
dsolve⁡ODE6
y⁡x=c__1⁢BesselJ⁡0,x+c__2⁢BesselY⁡0,x
newODE6 ≔ GenerateSimilarODE⁡ODE6
newODE6≔x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2−36⁢y⁡x=0
dsolve⁡newODE6
y⁡x=c__1⁢BesselJ⁡6,x+c__2⁢BesselY⁡6,x
ODE7 ≔ x2⁢%diff⁡y⁡x,x$2+x⁢%diff⁡y⁡x,x+x2−9⁢y⁡x=0
ODE7≔x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2−9⁢y⁡x=0
dsolve⁡ODE7
y⁡x=c__1⁢BesselJ⁡3,x+c__2⁢BesselY⁡3,x
newODE7 ≔ GenerateSimilarODE⁡ODE7
newODE7≔x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2−49⁢y⁡x=0
dsolve⁡newODE7
y⁡x=c__1⁢BesselJ⁡7,x+c__2⁢BesselY⁡7,x
ODEs that can be converted to a Bessel differential equation.
ODE8 ≔ x2⁢%diff⁡y⁡x,x$2+2⁢x⁢%diff⁡y⁡x,x+x2⁢y⁡x=0
ODE8≔x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+2⁢x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2⁢y⁡x=0
dsolve⁡ODE8
y⁡x=c__1⁢sin⁡xx+c__2⁢cos⁡xx
newODE8 ≔ GenerateSimilarODE⁡ODE8
newODE8≔x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+3⁢x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2−4⁢y⁡x=0
dsolve⁡newODE8
y⁡x=c__1⁢BesselJ⁡5,xx+c__2⁢BesselY⁡5,xx
ODE9 ≔ 2⁢x2⁢%diff⁡y⁡x,x$2+x⁢%diff⁡y⁡x,x+x2⁢y⁡x=0
ODE9≔2⁢x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2⁢y⁡x=0
dsolve⁡ODE9
y⁡x=c__1⁢x14⁢BesselJ⁡14,2⁢x2+c__2⁢x14⁢BesselY⁡14,2⁢x2
newODE9 ≔ GenerateSimilarODE⁡ODE9
newODE9≔5⁢x2⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+x2−4⁢y⁡x=0
dsolve⁡newODE9
y⁡x=c__1⁢x25⁢BesselJ⁡2⁢65,5⁢x5+c__2⁢x25⁢BesselY⁡2⁢65,5⁢x5
Terminating Laguerre polynomials.
ODE10 ≔ x⁢%diff⁡y⁡x,x$2+1−x⁢%diff⁡y⁡x,x+y⁡x=0
ODE10≔x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+1−x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+y⁡x=0
dsolve⁡ODE10
y⁡x=c__1⁢x−1+c__2⁢x−1⁢Ei1⁡−x+ⅇx
newODE10 ≔ GenerateSimilarODE⁡ODE10
newODE10≔x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+1−x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+2⁢y⁡x=0
dsolve⁡newODE10
y⁡x=c__1⁢x2−4⁢x+2+c__2⁢x2−4⁢x+2⁢Ei1⁡−x4+ⅇx⁢x−34
ODE11 ≔ x⁢%diff⁡y⁡x,x$2+1−x⁢%diff⁡y⁡x,x+5⁢y⁡x=0
ODE11≔x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+1−x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+5⁢y⁡x=0
dsolve⁡ODE11
y⁡x=c__1⁢x5−25⁢x4+200⁢x3−600⁢x2+600⁢x−120+c__2⁢x5−25⁢x4+200⁢x3−600⁢x2+600⁢x−120⁢Ei1⁡−x14400+ⅇx⁢x4−24⁢x3+177⁢x2−444⁢x+27414400
newODE11 ≔ GenerateSimilarODE⁡ODE11
newODE11≔x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x+1−x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x=0
dsolve⁡newODE11
y⁡x=c__1+Ei1⁡−x⁢c__2
Terminating Legendre polynomials.
ODE12 ≔ 1−x2⁢%diff⁡y⁡x,x$2−2⁢x⁢%diff⁡y⁡x,x+6⁢y⁡x=0
ODE12≔−x2+1⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−2⁢x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+6⁢y⁡x=0
dsolve⁡ODE12
y⁡x=c__1⁢−3⁢x2+1+c__2⁢3⁢x28−18⁢ln⁡x−1+−3⁢x28+18⁢ln⁡x+1+3⁢x4
newODE12 ≔ GenerateSimilarODE⁡ODE12
newODE12≔−x2+1⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−2⁢x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x−y⁡x−x2+1=0
dsolve⁡newODE12
y⁡x=c__1⁢x−x2+1+c__2−x2+1
ODE13 ≔ 1−x2⁢%diff⁡y⁡x,x$2−2⁢x⁢%diff⁡y⁡x,x+12⁢y⁡x=0
ODE13≔−x2+1⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−2⁢x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+12⁢y⁡x=0
dsolve⁡ODE13
y⁡x=c__1⁢−53⁢x3+x+c__2⁢−19+5⁢x3−3⁢x⁢ln⁡x−124+−5⁢x3+3⁢x⁢ln⁡x+124+5⁢x212
newODE13 ≔ GenerateSimilarODE⁡ODE13
newODE13≔−x2+1⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−2⁢x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+20⁢y⁡x=0
dsolve⁡newODE13
y⁡x=c__1⁢353⁢x4−10⁢x2+1+c__2⁢35⁢x4−30⁢x2+3⁢ln⁡x−1384+−35⁢x4+30⁢x2−3⁢ln⁡x+1384+35⁢x3192−55⁢x576
Chebyshev differential equation.
ODE14 ≔ 1−x2⁢%diff⁡y⁡x,x$2−x⁢%diff⁡y⁡x,x+25⁢y⁡x=0
ODE14≔−x2+1⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+25⁢y⁡x=0
dsolve⁡ODE14
y⁡x=c__1x+x2−15+c__2⁢x+x2−15
newODE14 ≔ GenerateSimilarODE⁡ODE14
newODE14≔−x2+1⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x,x−x⁢Typesetting:-_Hold⁡%diff⁡y⁡x,x+36⁢y⁡x=0
dsolve⁡newODE14
y⁡x=c__1x+x2−16+c__2⁢x+x2−16
The RandomTools[GenerateSimilarODE] command was introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
See Also
HowDoI,WorkWithRandomGenerators
InertForm
rand
RandomTools[Generate]
RandomTools[GenerateSimilar]
randpoly
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