# Documentation

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# `ode`::`polynomialSolutions`

Polynomial solutions of a homogeneous linear ordinary differential equation

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## Syntax

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

## Description

`ode::polynomialSolutions` computes a fundamental set of polynomial solutions of a homogeneous linear ordinary differential equation.

`ode::polynomialSolutions` returns a fundamental set of the polynomial solutions of `Ly`, i.e., solutions in the ring . When the option `Generic` is given, a generic form of them is returned.

## Examples

### Example 1

We compute the polynomial solutions of the following differential equation:

`Ly:=3*x*diff(y(x),x,x)-x*diff(y(x),x)+9*y(x)`

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

`ode::polynomialSolutions(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