E251 -- De integratione aequationis differentialis \(\frac{m dx}{\sqrt{}(1-x^4)} = \frac{n dy}{\sqrt{}(1-y^4)} \)
(On the integration of the differential equation \(\frac{m dx}{\sqrt{}(1-x^4)} = \frac{n dy}{\sqrt{}(1-y^4)} \))
Summary:
Euler takes it for granted that \(\frac{m}{n}\) is a rational number. In addition to the equation given in the title, Euler also handles the cases where there is an arbitrary whole fourth-degree function or a special 6th-degree function under the radical sign.
According to the records, it was presented to the St. Petersburg Academy on April 30, 1753.
Publication:
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Originally published in Novi Commentarii academiae scientiarum Petropolitanae 6, 1761, pp. 37-57
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Opera Omnia: Series 1, Volume 20, pp. 58 - 79
Documents Available:
- Original publication: E251
(in the Commentarii)
- English translation (Stacy Langton): E251
- German translation (Artur Diener and Alexander Aycock): E251
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E251 include:
- Dhombres J., “Some aspects of the history of functional-equations linked to the evolution of the function concept.” Archive for History of Exact Sciences, 36 (2), pp. 91-181 (1986).
- Landweber PS., “Elliptic genera - an introductory overview.” Lecture Notes in Mathematics, 1326, pp. 1-10 (1988).
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