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Paradoxical Euler: Integrating by Differentiating

Published online by Cambridge University Press:  23 January 2015

Andrew Fabian  and
Hieu D. Nguyen
Andrew Fabian
Affiliation:
Department of Mathematics, Rowan University, Glassboro, NJ 08028 USA, e-mails:Fabian78@Students.rowan.edu; Nguyen@rowan.edu
Hieu D. Nguyen
Affiliation:
Department of Mathematics, Rowan University, Glassboro, NJ 08028 USA, e-mails:Fabian78@Students.rowan.edu; Nguyen@rowan.edu
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Extract

Every student of calculus learns that one typically solves a differential equation by integrating it. However, as Euler showed in his 1758 paper (E236), Exposition de quelques paradoxes dans le calcul intégral (Explanation of certain paradoxes in integral calculus) [1], there are differential equations that can be solved by actually differentiating them again. This initially seems paradoxical or, as Euler describes it in the introduction of his paper:

Here I intend to explain a paradox in integral calculus that will seem rather strange: this is that we sometimes encounter differential equations in which it would seem very difficult to find the integrals by the rules of integral calculus yet are still easily found. not by the method of integration. but rather in differentiating the proposed equation again; so in these cases, a repeated differentiation leads us to the sought integral.

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 538 , March 2013 , pp. 61 - 74
Copyright
Copyright © The Mathematical Association 2013

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References

1. Euler, L., Exposition de quelques paradoxes dans le calcul intégral (Explanation of Certain Paradoxes in Integral Calculus), Mémoires de l'académie des sciences de Berlin 12 (1758) pp. 300–321; also published in Opera Omnia: Series 1, 22, pp. 214-236.Google Scholar
2. Fabian, A., English translation of [1] posted on the Euler Archive: http://www.eulerarchive.org Google Scholar
3. Sandifer, E., Curves and paradox, How Euler did it (MAA Column), October 2008: http://www.maa.org/news/howeulerdidit.html Google Scholar
4. Maclaurin, C., Geometria organica, sive descriptio linearum curvarum universalis, Oxford University (1720).Google Scholar
5. Anton, H., Bivens, I. and Davis, S., Calculus: early transcendental (8th edn.), John Wiley & Sons (2005).Google Scholar
6. Oprea, J., Differential geometry and its applications, MAA (2007).Google Scholar
7. Gray, A., Modern differential geometry of curves and surfaces with Mathematica (2nd edn.) CRC Press (1998).Google Scholar
8. Kühnel, W., Differential geometry: curves – surfaces – manifolds, (2nd edn.) (translated by Hunt, B.) American Mathematical Society, Student Mathematical Library, 16 (2006).Google Scholar