# Documentation

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# `ode`::`unimodular`

Unimodular transformation of a linear ordinary differential equation

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## Syntax

```ode::unimodular(`Ly`, y(`x`), <Transform>)
```

## Description

`ode::unimodular(Ly, y(x))` tests if the linear homogeneous differential equation `Ly` has a unimodular Galois group (i.e. the wronskian lies in the base field (x)), if not transforms `Ly` into a unimodular one (by changing the second highest coefficient to zero) and returns a `table` with index `equation` and `factorOfTransformation` containing respectively the transformed differential equation and the factor of transformation `Wn` such that a solution of the transformed equation multiplied by `Wn` is a solution of `Ly`.

If the option `Transform` is given then `Ly` is transformed unconditionally even if `Ly` has yet a unimodular Galois group.

## Examples

### Example 1

We test if the following differential equation has a unimodular Galois group:

` Ly := y(x)*6+x*diff(y(x),x)*(-2)+diff(y(x),x\$2)*(-x^2+1)`

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

It is unimodular since the factor of transformation is `1`. We can also check this by computing the wronskian of `Ly` which is a rational function:

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

Now we transform `Ly` into a differential equation whose wronskian is `1`:

`ode::unimodular(Ly, y(x), Transform)`

`ode::wronskian(%[equation], y(x))`

## Parameters

 `Ly` A homogeneous linear differential equation over ℚ(x). `y(x)` The dependent function of `Ly`.