Composition of Differential Operators

Question

For example, I have to write down the differential operator below as a polynomial in $\partial$ $$(\partial-f_n(x))(\partial-f_{n-1}(x))\cdots(\partial-f_1(x))$$where the product means composition.

Answer

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dx = Function[f, D[f, x]];
multiplyOp[scalar_, op_] := Function[{f1}, scalar op[f1]];
addOps[ops__] := Function[{f1}, Total@Map[#[f1] &, {ops}]];
CirclePlus[ops__] := addOps[ops];
CircleDot[scalar_, op_] := multiplyOp[scalar, op];
CircleTimes[ops__] := Composition[ops];
prod = (dx⊕f[2][x]⊙Identity)⊗(dx⊕f[1][x]⊙Identity);
CoefficientList[Simplify[prod[Exp[k x]]/Exp[k x]], k]

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Linear Algebra Done Right 3rd edition

What’s New in the Third Edition

  • This edition contains 561 exercises, including 337 new exercises that were not in the previous edition. Exercises now appear at the end of each section, rather than at the end of each chapter.
  • Many new examples have been added to illustrate the key ideas of linear algebra.
  • Beautiful new formatting, including the use of color, creates pages with an unusually pleasant appearance in both print and electronic versions.
  • Each theorem now has a descriptive name.

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Some Writings by Ziting

  • Jan 3 2014

2011年至2012年,我平淡无奇的生命中发生了一点类似“天花”的小小波折,可以算是目前为止人生中最大的转折点。

我像正常人一样,在之后的一段日子里焦虑和失眠,并伴有无意识的明显的情绪波动。

有幸遇到方程组,耳濡目染之下没有从一个极端走向另一个极端。

虽然现在小缺点因此变得一下子多了起来,但也还是摇摇摆摆希望做的更好。

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