generalized exponents of a linear homogeneous ODE

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DEtools[gen_exp] - generalized exponents of a linear homogeneous ODE

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

•

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

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

•

If the option restrict_to=S where is a subset of {minimal, integer, ramification1, rational}, then only a subset of the generalized exponents is given. If the option minimal is in , 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 (i.e. the generalized exponents in ) are given.

•	If a generalized exponent is a constant (if is in ) then 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 where in $(P)^(1) (=C union {infinity})$ is given, then this procedure first applies a transformation DEtools[translate] to move the point to the point , then computes the generalized exponents, and then substitutes in the result (or , if ). Note that this substitution only affects the part of the output that gives the relation between T and .

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

•	The argument domain describes the differential algebra. If this argument is the list , then the differential operators are notated with the symbols and . They are viewed as elements of the differential algebra where is the field of constants, and 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

(1)

(2)

>	$gen_exp(L, [Dx, x], T, x = infinity, 'restrict_to' = {'minimal'})$

(3)

>	$gen_exp(L, [Dx, x], T, x = infinity, 'restrict_to' = {'ramification1'})$

(4)

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

ode := diff(y(x), x, x)+y(x)/x^4 = 0

ode := diff(y(x), `$`(x, 2))+y(x)/x^4 = 0

(5)

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

(6)

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

(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:

ode := x^5*(diff(y(x), x, x, x))-x^2*(diff(y(x), x, x))+3*x*(diff(y(x), x))-15*y(x)*(1/4)

ode := x^5*(diff(y(x), `$`(x, 3)))-x^2*(diff(y(x), `$`(x, 2)))+3*x*(diff(y(x), x))-(15/4)*y(x)

(8)

[[T^(3/2)*(series(-1+(3/16)*T^2+(105/512)*T^4+O(T^6),T,6)), T^(3/2)*(series(T+(5/16)*T^3+(315/512)*T^5+O(T^6),T,6)), T = x], [T^3*exp(-(1/2)/T^2)*(series(1-(9/8)*T^2+(465/128)*T^4+O(T^6),T,6)), T = x]]

(9)

The above solutions can be rewritten as and where the dots refer to higher order terms. Note that because we took the point , and because the ramification index is in this example. Thus, the dominant terms are , and . Rewriting each of these in the form for some in one finds the following possible 's: , and . So those must be the generalized exponents of the ode at . Indeed:

(10)

	See Also
	DEtools/formal_sol, DEtools[indicialeq], diffop

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.