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# ode::exponents

Exponents of a linear ordinary differential equation

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```ode::exponents(Ly, y(x), p)
```

## Description

ode::exponents returns the set of exponents of a homogeneous linear differential equation at a given point.

ode::exponents(Ly, y(x), p) returns the set of (local) exponents of Ly at the place p. If the place is infinity then one uses instead. They are defined as roots (in an algebraic closure of (x)) of the indicial equation (c.f. ode::indicialEquation) of Ly so the set of exponents may be empty, see Example 2.

## Examples

### Example 1

We compute the exponents of the following differential equation at the regular point 0 and at the singular points -1 and infinity:

`Ly := diff(y(x),x\$2)+4/(x+1)*diff(y(x),x)+2/(x+1)^2*y(x)`

`ode::exponents(Ly, y(x), x)`

`ode::exponents(Ly, y(x), x+1)`

`ode::exponents(Ly, y(x), 1/x)`

### Example 2

It may happen that at a place the set of exponents is empty; this corresponds to an irregular singular point:

```Ly := (2*x+4)*diff(y(x),x)/(2*x+x^2-2)-2*y(x)/(2*x+x^2-2)-
(4*x+x^2)/(2*x+x^2-2)*diff(y(x),x\$2)+diff(y(x),x\$3)```

`ode::exponents(Ly, y(x), 1/x)`

`ode::exponents(Ly, y(x), x^2+2*x-2)`

## Parameters

 Ly A homogeneous linear differential equation over ℚ(x). y(x) The dependent function of Ly. p An irreducible polynomial in x or 1/x.

## Return Values

set, possibly empty.