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DEtools

  

muchange

  

change variables in the integrating factor of an ODE

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

muchange(tr, Mu, y(x), [t, u(t)])

Parameters

tr

-

set of transformation equations of the form {x=.., y(x)=..} from the old variables on the left hand side to the new variables on the right hand side

Mu

-

integrating factor of an ordinary differential equation (ODE)

y(x)

-

'dependent variable' of the problem (it can be any unknown function of one variable)

[t, u(t)]

-

(optional) new independent and dependent variables; required when they cannot be inferred from the transformation

Description

• 

The muchange command receives a change of variables, an integrating factor of an ODE, and the dependent variable y(x), and it returns the integrating factor of the problem in the new variables. The change of variables is performed by making calls to dchange, and hence the same extra arguments accepted by dchange are accepted by muchange as well.

• 

The change of variables performed by muchange is complementary to the (same) change of variables performed in the ODE in that, if μx,yx is an integrating factor of an ODE in y(x), then their product is a total derivative,

mu*ODE = Diff(R1(x,y(x)),x);

μODE=ⅆⅆxR1x,yx

(1)
  

and then, under a change of variables

tr := {x=X(t,u(t)), y(x)=Y(t,u(t))};

tr:=x=Xt,ut,yx=Yt,ut

(2)
  

where the new variables are {t, u(t)}, the following ODE is also exact:

'muchange(tr, mu, y(x)) * dchange(tr, ODE)' = Diff(R2(t,u(t)),t);

muchangetr,μ,yxdchangetr,ODE=ⅆⅆtR2t,ut

(3)
  

(note however that R1dchangetr,R2).

• 

This function is part of the DEtools package, and so it can be used in the form muchange(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[muchange](..).

Examples

withDEtools:

withPDEtools,dchange:

1. The following nonlinear second order ODE

ODE1ⅆ2ⅆx2yx+xⅆⅆxyx2+yxⅆⅆxyx+1+ⅆⅆxyxyxx=0

ODE1:=ⅆ2ⅆx2yx+xⅆⅆxyx2+yxⅆⅆxyx+1+ⅆⅆxyxyxx=0

(4)

has the following integrating factor.

Μ1intfactorODE1

Μ1:=xyx

(5)

This integrating factor can be tested using mutest:

mutestΜ1,ODE1

0

(6)

We now change variables

trx=1ut,yx=1t

tr:=x=1ut,yx=1t

(7)

so that the ODE becomes the following.

ODE2dchangetr,ODE1,normal

ODE2:=utt4ⅆⅆtut3+ut2ⅆⅆtut2t2+ⅆ2ⅆt2utut3tut2ⅆⅆtut2t+3ut3ⅆⅆtutt3ⅆⅆtut3=0

(8)

The integrating factor for this transformed ODE can be obtained from the integrating factor of ODE1 as follows:

Μ2muchangetr,Μ1,yx

Μ2:=ⅆⅆtutut3t

(9)

mutestΜ2,ODE2

0

(10)

2. muchange works as follows. Consider for instance a first order ODE turned exact by means of an integrating factor

exact_ODEμx,yxODEx,yx,ⅆⅆxyx=ⅆⅆxRx

exact_ODE:=μx,yxODEx,yx,ⅆⅆxyx=ⅆⅆxRx

(11)

and now consider the most general point transformation of variables:

trx=Xt,ut,yx=Yt,ut

tr:=x=Xt,ut,yx=Yt,ut

(12)

Perform a change of variables in the exact_ODE as a whole:

dchangetr,exact_ODE,t,ut,known=μ,ODE

μXt,ut,Yt,utODEXt,ut,Yt,ut,D1Yt,ut+D2Yt,utⅆⅆtutD1Xt,ut+D2Xt,utⅆⅆtut=D1Rt,ut+D2Rt,utⅆⅆtutD1Xt,ut+D2Xt,utⅆⅆtut

(13)

The left hand side in the above equation will also be an exact ODE if we multiply it by the denominator of the right hand side:

ⅆⅆtXt,ut

ⅆⅆtXt,ut

(14)

The new integrating factor for the transformed ODE is, generally speaking, given by

μ2t,ut=ⅆⅆtXt,ut'dchange''tr',μx,y

μ2t,ut=ⅆⅆtXt,utPDEtools:-dchangetr,μx,y

(15)

It is easy to see that this result is valid irrespective of the differential order of the ODE under consideration.

3. Consider the most general second order ODE having an integrating factor depending on (x, y'); this ODE is given by (see redode)

Μ1μx,ⅆⅆxyx

Μ1:=μx,ⅆⅆxyx

(16)

ODE1redodeΜ1,yx,2

ODE1:=ⅆ2ⅆx2yx=∫ⅆⅆxyxxμx,_aⅆ_a+D1_F1x,yx+D2_F1x,yxⅆⅆxyxμx,ⅆⅆxyx

(17)

What will be the integrating factor if we interchange the roles of the dependent and independent variables? The related transformation is given by

trx=ut,yx=t

tr:=x=ut,yx=t

(18)

The new integrating factor is

Μ2muchangetr,Μ1,yx,t,ut

Μ2:=ⅆⅆtutμut,1ⅆⅆtut

(19)

Transform the ODE and test the new integrating factor:

ODE2dchangetr,ODE1,t,ut,known=all,normal

ODE2:=ⅆ2ⅆt2utⅆⅆtut3=∫1ⅆⅆtutD1μut,_aⅆ_aⅆⅆtut+D1_F1ut,tⅆⅆtut+D2_F1ut,tⅆⅆtutμut,1ⅆⅆtut

(20)

mutestΜ2,ODE2

0

(21)

(Note that, when testing an integrating factor, mutest tests it against lhsODE2rhsODE2=0.)

See Also

DEtools[Lie]

DEtools[mutest]

dsolve/Lie

PDEtools[dchange]

 


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