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 $\mathrm{dvar}$ which is of the form $y\left(x\right)$.

An element $e$ in $C$ $\left[x^{-\frac{1}{r}}\right]$ is called a generalized exponent of L if there exists a formal solution $y$ of the form $y=s{\ⅇ}^{\int \frac{e}{x}\ⅆx}$ where $s$ is an element of $C$ $\left[\left[x^{\frac{1}{r}}\right]\right]$ $\left[\log \left(x\right)\right]$ 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 $C$ $\left[x^{-\frac{1}{r}}\right]$ then $r$ is called the ramification index of $e$.

The name T, which must be specified in the input, is used to denote $x^{-\frac{1}{r}}$ 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 $C$ $\left[\frac{1}{x}\right]$) 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 $P^1\left(=C\mathrm{union}\left\{\mathrm{infinity}\right\}\right)$ 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 $x=x-p$ in the result (or $x=\frac{1}{x}$, if $p=\mathrm{\infty }$). Note that this substitution only affects the part of the output that gives the relation between T and $x$.

The generalized exponents $e$ in $C$ $\left[x^{-\frac{1}{r}}\right]$ are computed only up to conjugation over the field $k\left(x\right)$, 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 $\left[\mathrm{Dx}\,x\right]$, then the differential operators are notated with the symbols $\mathrm{Dx}$ and $x$. They are viewed as elements of the differential algebra $C\left(x\right)$ $\left[\mathrm{Dx}\right]$ where $C$ is the field of constants, and $\mathrm{Dx}$ 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](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$L≔{\mathrm{Dx}}^5\+2x^3{\mathrm{Dx}}^3\+6x^2{\mathrm{Dx}}^2\+6x\mathrm{Dx}\+x^6\mathrm{Dx}\+1$

$L≔x^6\mathrm{Dx}+2x^3{\mathrm{Dx}}^3+{\mathrm{Dx}}^5+6x^2{\mathrm{Dx}}^2+6x\mathrm{Dx}+1$

$\mathrm{gen\_exp}\left(L\,\left[\mathrm{Dx}\,x\right]\,T\,x\=\mathrm{\∞}\right)$

$\left[\left[0\,T=\frac{1}{x}\right]\,\left[\frac{9}{4}+\frac{1}{T^5}\,\frac{11}{4}+\frac{1}{T^5}\,-T^2=\frac{1}{x}\right]\right]$

$\mathrm{gen\_exp}\left(L\,\left[\mathrm{Dx}\,x\right]\,T\,x\=\mathrm{\∞}\,\'\mathrm{restrict\_to}\'\=\left\{\'\mathrm{minimal}\'\right\}\right)$

$\left[\left[0\,T=\frac{1}{x}\right]\,\left[\frac{9}{4}+\frac{1}{T^5}\,-T^2=\frac{1}{x}\right]\right]$

$\mathrm{gen\_exp}\left(L\,\left[\mathrm{Dx}\,x\right]\,T\,x\=\mathrm{\∞}\,\'\mathrm{restrict\_to}\'\=\left\{\'\mathrm{ramification1}\'\right\}\right)$

$\left[\left[0\,T=\frac{1}{x}\right]\right]$

$\mathrm{ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\+\frac{y\left(x\right)}{x^4}\=0$

$\mathrm{ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{y\left(x\right)}{x^4}=0$

$\mathrm{gen\_exp}\left(\mathrm{ode}\,y\left(x\right)\,T\,x\=0\right)$

$\left[\left[\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)}{T}+1\,T=x\right]\right]$

$\mathrm{gen\_exp}\left(\mathrm{ode}\,y\left(x\right)\,T\,x\=0\,\'\mathrm{groundfield}\'\=\left[\mathrm{RootOf}\left(x^2\+1\right)\right]\right)$

$\left[\left[-\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)}{T}+1\,T=x\right]\,\left[\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)}{T}+1\,T=x\right]\right]$

$\mathrm{ode}≔x^5\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)-x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+3x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-\frac{15y\left(x\right)}{4}$

$\mathrm{ode}≔x^5\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)-x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+3x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-\frac{15y\left(x\right)}{4}$

$\mathrm{formal\_sol}\left(\mathrm{ode}\,y\left(x\right)\,T\,x\=0\right)$

$\left[\left[T^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(\mathrm{-1}+\frac{3}{16}{T}^2+\frac{105}{512}{T}^4+O\left(T^6\right)\right)\,T^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(T+\frac{5}{16}{T}^3+\frac{315}{512}{T}^5+O\left(T^6\right)\right)\,T=x\right]\,\left[T^3{\ⅇ}^{-\frac{1}{2T^2}}\left(1-\frac{9}{8}{T}^2+\frac{465}{128}{T}^4+O\left(T^6\right)\right)\,T=x\right]\right]$

$\mathrm{gen\_exp}\left(\mathrm{ode}\,y\left(x\right)\,T\,x\=0\right)$

$\left[\left[\frac{3}{2}\,\frac{5}{2}\,T=x\right]\,\left[\frac{1}{T^2}+3\,T=x\right]\right]$

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].