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`ode`::`dAlembert`

D'Alembert reduction of a linear homogeneous ordinary differential equation

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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.