The main functionalities of the LinearOperators package are the following.

- Given a linear equation with a d'Alembertian right-hand side, find a d'Alembertian solution if it exists.

- Given a d'Alembertian term, find a completely factorable annihilator of the term.

- Given a d'Alembertian term, find the minimal annihilator of the term.

- Given a d'Alembertian term, find the minimal completely factorable annihilator of the term.

- Given two operators, find their greatest common right divisor in factored form.

- Given an operator L, find the annihilator of the term g that is primitive for the solution f of Ly=0, and the operator K that converts f to g such that K(y)=g (if they exist). This is called accurate integration.

There are commands that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

Each command in the LinearOperators package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The long form, LinearOperators:-command, is always available. The short form can be used after loading the package.

The following is a list of available commands.

To display the help page for a particular LinearOperators command, see Getting Help with a Command in a Package.

$\mathrm{with}\left(\mathrm{LinearOperators}\right)$

$\left[\mathrm{Apply}\,\mathrm{DEToOrePoly}\,\mathrm{FactoredAnnihilator}\,\mathrm{FactoredGCRD}\,\mathrm{FactoredMinimalAnnihilator}\,\mathrm{FactoredOrePolyToDE}\,\mathrm{FactoredOrePolyToOrePoly}\,\mathrm{FactoredOrePolyToRE}\,\mathrm{IntegrateSols}\,\mathrm{MinimalAnnihilator}\,\mathrm{OrePolyToDE}\,\mathrm{OrePolyToRE}\,\mathrm{REToOrePoly}\,\mathrm{dAlembertianSolver}\right]$

$L≔\mathrm{OrePoly}\left(-x\,0\,1\right)$

$L≔\mathrm{OrePoly}\left(-x\,0\,1\right)$

$b≔\frac{\left(4x^3+1\right)\ln \left(x\right)}{x\mathrm{sqrt}\left(x\right)}$

$b≔\frac{\left(4x^3+1\right)\ln \left(x\right)}{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\mathrm{dAlembertianSolver}\left(L\,b\,x\,'\mathrm{differential}'\right)$

$-4\sqrt{x}\ln \left(x\right)$