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DEtools

  

ODEInvariants

  

computes relative invariants for linear and nonlinear ODEs of order 3 and higher

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

ODEInvariants(ODE, y(x))

Parameters

ODE

-

ordinary differential equation satisfied by y(x)

y(x)

-

(optional) dependent variable; required when the ODE contains more than one function being differentiated

Description

• 

Given a linear or nonlinear ODE of order m=3 or higher, ODEInvariants returns a list of m2 relative invariants under transformations of the formxFx,yxPxyx. The weight of each of these relative invariants is given by the power of the derivative of Fx entering as a factor in the transformed invariant, and given two relative invariants Ir and Is respectively of weights r and s, an absolute invariant can be constructed by taking IrsIsr (see references [1] and [2]).

• 

The invariants in the returned list are ordered according to increasing weight, from weight = 3 to weight = m, the order of the equation. For example, for a fourth order ODE, the returned list contains two relative invariants, respectively of weights 3 and 4.

• 

In the case of linear ODEs, these invariants coincide with the Wilczynski invariants (see reference [3]) although their computation is performed without rewriting the linear equation in Laguerre-Forsyth form. Instead, given a linear ODE of order 3 or higher, in normal form,

           ym+c[m2]xym2+...+c[1]xy' +c[0]xy=0

(1)

  

by transforming this equation using

xFx,yF'm12ut

  

we obtain an equation of the same form as (1). Performing now a sequential reduction of the transformed c[mj]x coefficients, j=3..m, eliminating derivatives of Fx, a sequence of expressions result that coincide with the Wilczynski relative invariants. The advantage of this process if that it does not require rewriting the linear equation in Laguerre-Forsyth form, which in turn would require solving a linear ODE of order m-1.

• 

For nonlinear ODEs of order m=3 or higher, that are polynomial in the unknown yx and its derivatives, an auxiliary linear ODE is constructed - say in ux - where the coefficient of each derivative of ux in this linear ODE is equal to the coefficient of the derivative of yx of the same order in the given nonlinear ODE. Thus, because the ODE in yx is nonlinear, this auxiliary linear ODE in ux has coefficients involving yx and its derivatives. Next the Wilczynski invariants are computed for this linear ODE in ux and finally they are reduced with respect to the given nonlinear ODE in yx (i.e., the mth derivative of yx is isolated and replaced in the invariants).

• 

Note that in the nonlinear case the invariants may dependent on the unknown yx and its derivatives. However, if the nonlinear equation is linearizable through a point transformation these invariants will depend only on the independent variable x - see examples below.

Examples

withDEtools,ODEInvariants

ODEInvariants

(1)

PDEtools:-declareC,F,c,u,yx,prime=x

Cxwill now be displayed asC

Fxwill now be displayed asF

cxwill now be displayed asc

uxwill now be displayed asu

yxwill now be displayed asy

derivatives with respect toxof functions of one variable will now be displayed with '

(2)

Consider the general form of a third order linear ODE

ode3ⅆ3ⅆx3yx=addc[j]xⅆjⅆxjyx,j=2,1,0

ode3:=y'''=yc0+y'c1+y''c2

(3)

For ODEs of third order ODEInvariants returns one invariant

ODEInvariantsode3

2c0+23c1c2+427c23+13c2''c1`'`23c2`'`c2

(4)

Let's check that the returned invariants are relative invariants in the case of a  fourth order linear ODE

ode4ⅆ4ⅆx4yx=addc[j]xⅆjⅆxjyx,j=3,2,1,0

ode4:=y''''=yc0+y'c1+y''c2+y'''c3

(5)

iiODEInvariantsode4

ii:=c1+12c2c3+18c33+12c3''c2`'`34c3c3`'`,14c3'''c2''34c3''c33340c3`'`2+1320312c32+432c2c3`'`39320c341320c32c254c1c3920c22+54c2`'`c35c0+52c1`'`

(6)

By definition, these expressions are relative invariants if when we transform in them the coefficients c[j](x) using

trx=Ft,yx=ⅆⅆtFt32ut

tr:=x=Ft,y=Ft3/2ut

(7)

the resulting expressions are of the form F`' `kΦcjF, and if next, by replacing F by the identity, we reobtain the departing expressions ii

So we proceed first transforming these coefficients entering ii and for that purpose transform ode[4]

PDEtools:-dchangetr,ode4,t,ut,known=all,simplify

116135utFt,t4300utFt,t2Ft,t,tFt+120utFt,tFt,t,t,tFt2+60utFt,t,t2Ft224utFt,t,t,t,tFt3240Ft,t3utFt+120Ft,t2ut,tFt2+320Ft,tutFt,t,tFt280utFt,t,t,tFt380Ft,t,tut,tFt316ut,t,t,tFt4Ft13/2=188c0FtutFt6+12c1FtutFt,tFt4+8c1FtutFt56c2FtutFt,t2Ft2+12c2FtutFt,t,tFt3+16c2FtFt,tutFt3+8c2Ftut,tFt4+15c3FtutFt,t330c3FtutFt,tFt,t,tFt+12c3FtutFt,t,t,tFt230c3FtFt,t2utFt+12c3FtFt,tut,tFt2+28c3FtutFt,t,tFt2+8c3Ftut,t,tFt3Ft9/2

(8)

To get the transformed coefficients C[j]x, first isolate u''''

subst=x,isolate,ⅆ4ⅆt4ut

u''''=1162F'28c0FuF'6+12c1FuF''F'4+8c1Fu'F'56c2FuF''2F'2+12c2FuF'''F'3+16c2FF''u'F'3+8c2Fu''F'4+15c3FuF''330c3FuF''F'''F'+12c3FuF''''F'230c3FF''2u'F'+12c3FF''u''F'2+28c3Fu'F'''F'2+8c3Fu'''F'3135uF''4+300uF''2F'''F'120uF''F''''F'260uF'''2F'2+24uF'''''F'3+240F''3u'F'120F''2u''F'2320F''u'F'''F'2+80u'F''''F'3+80F'''u''F'3F'4

(9)

Compute now the coefficients C[j]x of derivatives of ux in the transformed equation

zip`=`,C[3]x,C[2]x,C[1]x,C[0]x,PDEtools:-dcoeffsrhs,ux

C3=F'c3F,C2=F'2c2F+32F''c3F5F'''F'+152F''2F'2,C1=F'3c1F+2F'F''c2F+72F'''c3F154F''2c3FF'5F''''F'+20F''F'''F'215F''3F'3,C0=F'4c0F+32F'2c1FF''+32F'c2FF'''34F''2c2F+32c3FF''''154F''F'''c3FF'+158F''3c3FF'232F'''''F'+152F''F''''F'2+154F'''2F'2754F''2F'''F'3+13516F''4F'4

(10)

Compute now the invariants ii using these coefficients C[j]x expressed in terms of the c[j]x using the formula above

subsc=C,ii

C1+12C2C3+18C33+12C3''C2`'`34C3C3`'`,14C3'''C2''34C3''C33340C3`'`2+1320312C32+432C2C3`'`39320C341320C32C254C1C3920C22+54C2`'`C35C0+52C1`'`

(11)

factoreval,

18F'3c3F36Dc3Fc3F+4c2Fc3F+4D2c3F8Dc2F+8c1F,1320F'439c3F4+312Dc3Fc3F2208c2Fc3F2240D2c3Fc3F+400Dc2Fc3F264Dc3F2+432Dc3Fc2F400c1Fc3F144c2F2+800Dc1F+80D3c3F320D2c2F1600c0F

(12)

It is visible that each expression is now of the form F`' `kΦcjF, and according to the description, the first relative invariant has weight 3 (in the factor F' k,k=3) and the second one has weight 4. Let's verify that at F=identity we reobtain the departing expressions ii, proving in that way that the expressions ii are relative invariants

convertF=x→x|F=x→x,diff

c1+12c2c3+18c33+12c3''c2`'`34c3c3`'`,39320c34+3940c3`'`c321320c32c234c3''c354c1c3+54c2`'`c33340c3`'`2+2720c3`'`c2920c22+14c3'''+52c1`'`5c0c2''

(13)

normalii

0,0

(14)

Let's now transform the linear equation ode[4] into a nonlinear one by means of a point transformation

PDEtools:-dchangeyx=1ux,ode4,ux

24u'4u536u'2u''u4+6u''2u3+8u'u'''u3u''''u2=c36u'3u4+6u'u''u3u'''u2+c22u'2u3u''u2c1u'u2+c0u

(15)

nonlinearODEisolate,ⅆ4ⅆx4ux

nonlinearODE:=u''''=c36u'3u4+6u'u''u3u'''u2+c22u'2u3u''u2c1u'u2+c0u24u'4u5+36u'2u''u46u''2u38u'u'''u3u2

(16)

ODEInvariantsnonlinearODE

c1+12c2c3+18c33+12c3''c2`'`34c3c3`'`,14c3'''c2''34c3''c33340c3`'`2+1320312c32+432c2c3`'`39320c341320c32c254c1c3920c22+54c2`'`c35c0+52c1`'`

(17)

The expressions above depend only on x, not on ux or its derivatives, because this nonlinear ODE above is related - by construction - to a linear ODE (ode[4]) through a point transformation (y1y used above). Moreover: the invariants are the same as those in ii, of the related linear ode[4]. When the nonlinear ODE cannot be related to a linear ODE through a point transformation, the invariants depend on the dependent variable and perhaps also its derivatives. For example:

ⅆ4ⅆx4ux=ux3ⅆⅆxux+ⅆ2ⅆx2ux2+x

u''''=u3u'+u''2+x

(18)

ODEInvariants

u32u''',152u'u2195u''22u3u'2x

(19)

References

  

[1] Olver, P.J. Equivalence, Invariants and Symmetry. Cambridge Press, 1995.

  

[2] Chalkley, R., Basic Global Relative Invariants for Homogeneous Linear Differential Equations, Amer Mathematical Society (2002).

  

[3] Wilczynski, E.J., Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1905.

See Also

dchange

dcoeffs

declare

DEtools

PDEtools

ReducedForm

 


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