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# ode::dAlembert

D'Alembert reduction of a linear homogeneous ordinary differential equation

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```ode::dAlembert(Ly, y(x), v)
```

## Description

ode::dAlembert(Ly, y(x), v) returns the reduced differential equation of Ly using the method of reduction of d'Alembert and the function v. If v is a solution of Ly and u is a solution of the reduced differential equation then vtu is another solution of Ly.

## Examples

### Example 1

Consider the following differential equation:

```Ly := 2/x^3*y(x)-2/x^2*diff(y(x),x)+1/x*diff(y(x),x\$2)+
diff(y(x),x\$3)```

We easily check that x is a particular solution of Ly:

`ode::evalOde(Ly, y(x)=x)`

Then we reduce the equation Ly using this special solution:

`R := ode::dAlembert(Ly, y(x), x)`

The solutions of the equation R are not too hard to find:

`ode::evalOde(R, y(x)=1), ode::evalOde(R, y(x)=1/x^3)`

So a basis of solutions of Ly is therefore which can be checked directly:

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

## Parameters

 Ly A homogeneous linear differential equation. y(x) The dependent function of Ly. v An expression.

Expression.