Linear Differential Operators - Maple Programming Help

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Linear Differential Operators

In Maple, one can work with differential operators by using a subset of the DEtools package. Differential operators in this case are polynomial objects of the form



with the coefficients ${a}_{i}\left(x\right)$ being rational functions over a field. Here $\mathrm{D}$ is an object that satisfies  properties such as $\mathrm{DF}\left(x\right)=1$ and $\mathrm{DF}\left(uv\right)=u\mathrm{DF}\left(v\right)+\mathrm{DF}\left(u\right)v$.  One can perform functions on these operators such as multiplying, forming symmetric products, taking one-sided greatest common divisors, factoring, and so on.

This functionality, due to Mark van Hoeij (University of Nijmegen), allows for significant future advances for applications such as finding closed form solutions of linear differential equations.  Some applications of this subpackage of DEtools are included below.

We use the with command to allow us to use the short form for the commands in the DEtools package.

 > $\mathrm{restart}$
 > $\mathrm{with}\left(\mathrm{DEtools}\right):$