# Documentation

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# `ode`::`isFuchsian`

Tests if a homogeneous linear ordinary differential equation is of Fuchsian type

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## Syntax

```ode::isFuchsian(`Ly`, y(`x`), <AllExponents>)
```

## Description

`ode::isFuchsian` returns `TRUE` if `Ly` is of Fuchsian type, i.e., all the singular points (including the point at infinity) of `Ly` are regular. It returns `FALSE` if at least one singular point is irregular. When the option `AllExponents` is given, either `FALSE` is returned or a list where each element is a table containing, at each regular singular point of `Ly` the place, the indicial equation and the exponents.

## Examples

### Example 1

We test if the following differential equation is Fuchsian:

`Ly:=x*(1-x)*diff(y(x),x\$2)+(1-x)*diff(y(x),x)+10*y(x)`

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

We can have a look of the indicial equations, exponents at each regular singular point of `Ly`:

`ode::isFuchsian(Ly, y(x), AllExponents)`

### Example 2

In this example, the Airy equation, the only singular point is at infinity and is irregular:

`ode::isFuchsian(diff(y(x),x\$2)-x*y(x), y(x))`

## Parameters

 `Ly` A homogeneous linear ordinary differential equation with coefficients in the field ℚ(x) of rational functions over the rationals. `y(x)` The dependent function of `Ly`.

## Options

 `AllExponents` Return a list of tables of indical equations and exponents for regular singular points.