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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.

Return Values

Expression.

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