# Documentation

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# `ode`::`companionSystem`

Companion matrix of a linear homogeneous ordinary differential equation

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## Syntax

```ode::companionSystem(`Ly`, y(`x`), <`R`>)
```

## Description

`ode::companionSystem(Ly, y(x))` returns the companion matrix associated to `Ly`. If the optional argument `R` is given, the elements of the matrix are in `R`.

## Examples

### Example 1

We compute the companion matrix of the following differential equation:

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

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

## Parameters

 `Ly` A linear homogeneous ordinary differential equation. `y(x)` The dependent function of `Ly`. `R` A field of functions or numbers of characteristic zero, default is `Dom::ExpressionField(normal)`.

## Return Values

Object of type `Dom::Matrix`.