# Documentation

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# `ode`::`ratSys`

Rational solutions of a first order homogeneous linear differential system

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## Syntax

```ode::ratSys(`M`, `x`)
```

## Description

`ode::ratSys(M, x)` computes a fundamental set of rational solutions of the first order homogeneous linear differential system `Y'=MY`. This method uses a cyclic vector and therefore is not optimal.

## Examples

### Example 1

We compute the rational solutions of the following differential system:

```A := matrix([ [2*(x+x^2-9)/x/(x-2),2*(x^2-6)/x/(x-2)], [-3*(2*x+x^2-12)/x/(x-2),-(2*x+3*x^2-24)/x/(x-2)] ])```

`v := ode::ratSys(A, x)`

And we can check the result:

`diff(v[1], x) = normal(A*v[1])`

## Parameters

 `M` A square matrix of type `Dom::Matrix` with coefficients in the field ℚ(x) of rational functions over the rationals. `x` The independent function.

## Return Values

`set`, possibly empty, of objects of type `Dom::Matrix`.