# Documentation

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# `ode`::`scalarEquation`

Transforms a linear differential system to an equivalent scalar linear differential equation

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## Syntax

```ode::scalarEquation(`A`, `x`, `y`, <Transform>)
```

## Description

`ode::scalarEquation` converts a first order homogeneous linear differential system to an equivalent homogeneous scalar linear differential equation using the method of cyclic vector.

`ode::scalarEquation(A, x, y)` returns a scalar homogeneous linear differential equation in `y(x)` equivalent to the first order homogeneous differential system using the method of the cyclic vector. If the option `Transform` is given then a list is returned whose first element is the corresponding differential equation `Ly` and second element is an invertible matrix `P` such that is the companion matrix associated to `Ly`; hence if `Z` is a solution of the differential system then PZ is a solution of the system .

## Examples

### Example 1

We compute a linear differential equation equivalent to the following differential system:

`A := matrix( [ [x^2-1,1,0], [0,x^2+5*x+1/3,1], [0,0,2]])`

`l := ode::scalarEquation(A, x, y, Transform)`

And we can check that, for `P=l[2]`, is the companion matrix associated to `l[1]`:

```P := l[2]: bool( diff(P,x)*P^(-1)+P*A*P^(-1) = ode::companionSystem(l[1], y(x)) )```

## Parameters

 `A` A square matrix of type `Dom::Matrix`. `x` The independent variable of the resulting scalar differential equation. `y` The dependent variable of the resulting scalar differential equation.

## Return Values

Expression or a list.