Example of the construction of equations
In this paper, Euler investigates a differential equation that he encountered in finding the arc length of an ellipse. This differential equation cannot be solved by separation of variables, as is indicated in the title of the article. Euler first develops a formula for the arc length of an ellipse by cleverly manipulating a binomial series, then shows that this formula satisfies the desired differential equation. Integrating factors make a brief appearance.
Original Source Citation
Commentarii academiae scientiarum Petropolitanae, Volume 6, pp. 168-174.
Opera Omnia Citation
Series 1, Volume 20, pp.1-7.