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This MSE post asked about a specific integration technique that appears to be attributed to Charles Hermite, per a comment. The OP's source calls the technique el método alemán, i.e. the German method.

As far as I can tell, this is just an application of finding undetermined coefficients (UC), which as the link indicates is most often a topic covered by introductory courses on ODEs, but it's also a component of partial fraction decomposition, and it can be used to rewrite a polynomial $p(x)$ to another polynomial $q(x-c)$ which can help in polynomial division.

Considering the frequent use of the German word ansatz in the context of UC, I wonder if this is the association that is being made by the Spanish name for Hermite's method.

I have two questions.

  1. Do we know who is credited for coming up with the method of UC? Is any single person, German or otherwise, known for being the first to employ this method in any application?

I cannot find any actual history behind the method. Almost everything I've come across has essentially been lecture notes, chapters from textbooks, or video lessons that simply introduce and/or demonstrate the method.

While composing this question, I found an older HSM.SE post that credits Euler with a particular ansatz to solve linear, constant-coefficient (LCC) ODEs.

I also came across a set of lecture notes mentioning (German mathematician!) Ernst Kummer which I believe was a false lead, as the author's use of "Kümmer's method" seems to refer to transforming an ODE into LCC form.

  1. Is there any known history behind the name, German method ?

I've tried searching Google with "método alemán" integral -... before:... after:... where I use multiple omissions -... to filter out certain disciplines (the string appears to be a common match in several articles about hydrology and civil engineering, for example) and the time range to make parsing the hits a bit easier on the eyes. No luck here, either.

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  • $\begingroup$ My first guess was to look in Differential Equations with Applications and Historical Notes by George G. Simmons, but I couldn't find any historical notes on this method in his book (but I didn't look very hard either). If no one answers in the next day or so, I'll look more carefully through the many mathematics history books on my shelves and see if anything turns up. Of course, if you're at a university, you can simply look at such books in your university's library . . . $\endgroup$ Commented Apr 3 at 19:16
  • $\begingroup$ @DaveLRenfro Sadly, no uni library access. I appreciate the effort and reference, though! Pleased to see an author take some time to put faces to names, so to speak. $\endgroup$
    – user170231
    Commented Apr 3 at 19:34
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    $\begingroup$ Katz attributes the method of undetermined coefficients to Leibniz. Probst writes in the Oxford Handbook, p. 218:"Another new approach was using the method of undetermined coefficients with infinite series characterizing transcendent curves in finding solutions to certain inhomogeneous differential equations (Supplementum geometriae practicae; GM V 285– 288)." Supplementum is from 1693. $\endgroup$
    – Conifold
    Commented Apr 4 at 1:06
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    $\begingroup$ However, Newton encrypted the idea earlier in his famous anagram from a 1676 letter to Leibniz:"One method consists in extracting a fluent quantity from an equation at the same time involving its fluxion; but another by assuming a series for any unknown quantity whatever, from which the rest could conveniently be derived, and in collecting homologous terms of the resulting equation in order to elicit the terms of the assumed series." Poincare later 'translated' it to say:"I know how to integrate all differential equations." $\endgroup$
    – Conifold
    Commented Apr 4 at 1:20
  • $\begingroup$ @Conifold Very interesting, thanks. Leibniz would certainly fit the bill for German, and I wouldn't be surprised if he was largely or partly responsible for popularizing UC. On the other hand, I think I did manage to find a solid lead re. Hermite's involvement with the aforementioned integration technique. Perhaps his work was translated at some point into German, then Spanish, and spread from there? Unsure if this kind of dissemination was likely at the time (1870s?). $\endgroup$
    – user170231
    Commented Apr 4 at 15:03

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The method of undetermined coefficients was probably first developed by Euler, he applied it in his work to analyze the perturbations of Saturn by Jupiter for the Paris Academy's prize competition of 1748. (Euler was awarded the prize.) He solved the relevant differential equations by the method of undetermined coefficients, and used multiple equations of condition, based on observations, to effect the differential correction of orbital elements and to fix coefficients of some perturbational terms.

Shortly afterwards, or perhaps partly in simultaneity, an essentially similar method was used by Alexis-Claude Clairaut to solve differential equations defining the moon's orbital motion in longitude, in a work "Theorie de la Lune" which was considered in another prize competition and then awarded the prize, this time by the St Petersburg Academy of Sciences, which then published the work.

A general description of both works is offered in Curtis Wilson's "Factoring the lunar problem: geometry, dynamics, and algebra in the lunar theory from Kepler to Clairaut", p.39-58 in "Hamiltonian Dynamical Systems: History, Theory and Applications", eds K M Dumas et al., Springer, 1995. Clairaut's "Theorie de la Lune" is available onlne at (https://gallica.bnf.fr/ark:/12148/bpt6k110454w/f1.item.texteImage).

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