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Student[ODEs][ReduceOrder]

  

LinearParticularSolution

  

Reduce the order of a linear ODE using a particular solution

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

LinearParticularSolution(ODE, y(x) = psol)

LinearParticularSolution(ODE, y(x) = psol, u(t))

Parameters

ODE

-

second order linear ordinary differential equation

y(x) = psol

-

equation; a particular solution of the ODE

y

-

name; the original dependent variable

x

-

name; the original independent variable

u

-

name; the dependent variable for the reduced ODE

t

-

name; the independent variable for the reduced ODE

Description

• 

The LinearParticularSolution(ODE, psol) command reduces the order of a second order linear ODE using a particular solution.

• 

The second argument must be a particular solution of the ODE of the form y(x) = psol, where psol does not depend on y(x).

• 

The third argument, u(t), representing the variable for the reduced ODE, is optional. If it is not given, new independent and dependent variables will be chosen which do not conflict with the existing variables.

• 

The default output is a sequence consisting of the reduced ODE in terms of the new variables, followed by the transformation used to recover the original ODE from the reduced ODE.

• 

If an extra option solve or solve=true is also given, an attempt is made to solve the reduced ODE and return the general solution to the original ODE. If successful, the general solution of the original ODE will be returned.

• 

If the option solve is given and furthermore, the extra option output=basis is given, then as above an attempt will be made to find the general solution to the original ODE, but the answer will be returned in the form of a list of particular solutions forming a basis for that general solution.

Examples

withStudentODEs:

withStudentODEsReduceOrder:

Given a linear homogeneous ODE:

odex1diffyx,x,xxdiffyx,x+yx=0

odex1ⅆ2ⅆx2yxxⅆⅆxyx+yx=0

(1)

And a particular solution for it:

psolyx=expx

psolyx=ⅇx

(2)

Use the particular solution to find a new ODE of reduced order which can be used to solve the original:

reduced_ode,trLinearParticularSolutionode,psol

reduced_ode,trⅆⅆtut=utt2t1,t=x,ut=ⅆⅆxyxⅇx

(3)

Solve the reduced order ODE:

reduced_solSolvereduced_ode,ut

reduced_solut=ⅇt+_C1t1

(4)

Apply the transformation to find a simpler ODE for the original variable y(x):

new_odeevalreduced_sol,tr

new_odeⅆⅆxyxⅇxyxⅇx=ⅇx+_C1x1

(5)

Solve the simpler ODE to find the general solution of the original ODE:

gensol1Solvenew_ode,yx

gensol1yx=ⅇxxⅇx+_C1_C2

(6)

Alternatively, the particular solution can be used to solve the original ODE in a single step if we use the option solve:

gensolLinearParticularSolutionode,psol,solve

gensolyx=ⅇxxⅇx+_C1+_C2

(7)

Simplify the form of the general solution:

simplifyexpandgensol

yx=xⅇ_C1+ⅇx_C2

(8)

Use the output=basis option instead:

BasisLinearParticularSolutionode,psol,solve,output=basis

Basisx,ⅇx

(9)

Show the new solution:

solyx=remove`=`,Basis,rhspsol1

solyx=x

(10)

Calculate the Wronskian of the two solutions for y(x):

WVectorCalculus:-WronskianBasis,x

Wxⅇx1ⅇx

(11)

The determinant should be nonzero to prove independence of the solutions:

simplifyLinearAlgebra:-DeterminantW

ⅇxx1

(12)

Compatibility

• 

The Student[ODEs][ReduceOrder][LinearParticularSolution] command was introduced in Maple 2021.

• 

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

See Also

dsolve

Student

Student[ODEs]

Student[ODEs][ReduceOrder]