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# ode::scalarEquation

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

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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.