# Documentation

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# `ode`::`rationalSolutions`

Rational solutions of a homogeneous linear ordinary differential equation

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## Syntax

```ode::rationalSolutions(`Ly`, y(`x`), <Generic>)
```

## Description

`ode::rationalSolutions` returns a fundamental set of the rational solutions of `Ly`, i.e., solutions in the field (x). When the option `Generic` is given, a generic form of them is returned.

## Examples

### Example 1

We compute the rational solutions of the following differential equation:

```Ly:=(4*x^5+8*x^3+4*x)*diff(y(x),x,x)+ (36*x^4+32*x^2-4)*diff(y(x),x)+48*x^3*y(x)```

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

`ode::rationalSolutions(Ly, y(x), Generic)`

## 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`.

## Return Values

`set`, possibly empty, of functions or an expression