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gen_exp

  

generalized exponents of a linear homogeneous ODE

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

gen_exp(L, domain, T, opt)

gen_exp(eqn, dvar, T, opt)

Parameters

L

-

differential operator

domain

-

list containing two names

T

-

name

opt

-

(optional) sequence of options

eqn

-

homogeneous linear differential equation

dvar

-

dependent variable

Description

• 

The input is a differential operator L or a linear ODE (ordinary differential equation) eqn having rational function coefficients.

• 

The output is a list of lists. Each of these lists contains one equivalence class of generalized exponents.

• 

Let x be the independent variable. If a differential operator is specified, then x is the second element of the list domain. If an ODE is specified, then x is implicitly given in dvar which is of the form yx.

• 

An element e in C x1r is called a generalized exponent of L if there exists a formal solution y of the form y=sⅇexⅆx where s is an element of C x1r logx with valuation 0, which means that the coefficient of 1 in s is not zero, for more details see the help page of formal_sol. If r is the smallest positive integer for which e is in C x1r then r is called the ramification index of e.

• 

The name T, which must be specified in the input, is used to denote x1r times a constant. This procedure computes the generalized exponents and expresses them in terms of T. The relation between T and x is given in the output as well, in each equivalence class of generalized exponents.

• 

If the option restrict_to=S where S is a subset of {minimal, integer, ramification1, rational}, then only a subset of the generalized exponents is given. If the option minimal is in S, then only the minimal generalized exponent in each equivalence class will be given. If the option integer or rational is given then only the generalized exponents in Z or Q, respectively, are given. If the option ramification1 is given, then only the generalized exponents with ramification index 1 (i.e. the generalized exponents in C 1x) are given.

• 

If a generalized exponent e is a constant (if e is in C) then e is an exponent. The exponents are the solutions of the indicial equation. If not all generalized exponents are constants, then the ODE is called irregular singular.

• 

If the optional argument x=p where p in P1=Cunion{infinity} is given, then this procedure first applies a transformation DEtools[translate] to move the point p to the point 0, then computes the generalized exponents, and then substitutes x=xp in the result (or x=1x, if p=). Note that this substitution only affects the part of the output that gives the relation between T and x.

• 

The generalized exponents e in C x1r are computed only up to conjugation over the field kx, where k is the minimal field of constants over which the input is defined. A larger field k can be specified by the option groundfield = list of RootOfs.

• 

The argument domain describes the differential algebra. If this argument is the list Dx,x, then the differential operators are notated with the symbols Dx and x. They are viewed as elements of the differential algebra Cx Dx where C is the field of constants, and Dx denotes the differentiation operator.

• 

If the argument domain is omitted then the differential algebra specified by the environment variable _Envdiffopdomain is used. If this environment variable is not set then the argument domain may not be omitted.

• 

Instead of a differential operator, the input can also be a linear homogeneous ODE having rational function coefficients. In this case, the second argument dvar must be the dependent variable.

• 

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

Examples

withDEtools:

LDx5+2x3Dx3+6x2Dx2+6xDx+x6Dx+1

Lx6Dx+2x3Dx3+Dx5+6x2Dx2+6xDx+1

(1)

gen_expL,Dx,x,T,x=∞

0,T=1x,94+1T5,114+1T5,T2=1x

(2)

gen_expL,Dx,x,T,x=∞,'restrict_to'='minimal'

0,T=1x,94+1T5,T2=1x

(3)

gen_expL,Dx,x,T,x=∞,'restrict_to'='ramification1'

0,T=1x

(4)

Note: The quotes around the names in the options may be omitted unless a value has been assigned to those names.

odeⅆ2ⅆx2yx+yxx4=0

odeⅆ2ⅆx2yx+yxx4=0

(5)

In this example the field of definition is Q, so the generalized exponents at x=0 will be given up to conjugation over Q:

gen_expode,yx,T,x=0

RootOf_Z2+1T+1,T=x

(6)

Now specify the field Qi. Since both generalized exponents are defined over this field, both will appear in the output:

gen_expode,yx,T,x=0,'groundfield'=RootOfx2+1

RootOf_Z2+1T+1,T=x,RootOf_Z2+1T+1,T=x

(7)

Each generalized exponent gives the dominant term (ignoring logarithmic factors) in a formal solution. For example, consider the formal solutions of the following ode:

odex5ⅆ3ⅆx3yxx2ⅆ2ⅆx2yx+3xⅆⅆxyx15yx4

odex5ⅆ3ⅆx3yxx2ⅆ2ⅆx2yx+3xⅆⅆxyx15yx4

(8)

formal_solode,yx,T,x=0

T32−1+316T2+105512T4+OT6,T32T+516T3+315512T5+OT6,T=x,T3ⅇ12T2198T2+465128T4+OT6,T=x

(9)

The above solutions can be rewritten as T32constant+...,T52constant+... and T3ⅇ12T2constant+... where the dots refer to higher order terms. Note that T=x because we took the point x=0, and because the ramification index is r=1 in this example. Thus, the dominant terms are x32,x52, and x3ⅇ12x2. Rewriting each of these in the form ⅇexⅆx for some e in C x1r one finds the following possible e's: 32,52, and 3+1x2. So those must be the generalized exponents of the ode at x=0. Indeed:

gen_expode,yx,T,x=0

32,52,T=x,1T2+3,T=x

(10)

References

  

Cluzeau, T., and van Hoeij, M. "A Modular Algorithm to Compute the Exponential Solutions of a Linear Differential Operator." J. Symb. Comput. Vol. 38, 2004: 1043-1076.

  

Ince, E.L. Ordinary Differential Equations, Chap. XVI-XVII. New York: Dover Publications, 1956.

  

van der Put, M., and Singer, M. F. Galois Theory of Linear Differential Equations, Vol. 328. Springer: 2003. An electronic version of this book is available at http://www4.ncsu.edu/~singer/ms_papers.html.

  

* More information on generalized exponents is in the help page for DEtools[formal_sol].

See Also

DEtools/formal_sol

DEtools[indicialeq]

diffop