In his first paper on Elliptic Integrals, Euler calculates the arc length of an ellipse by solving a non-separable differential equation arising from it. He previously thought no non-separable differential equation could be solved. He says:
This case seemed exceedingly paradoxical to me at first; but after studying the solution more carefully I realized easily not only that a separation could not be deduced from it, but also, that if a separation were to succeed by another method, far greater absurdities would follow. One might find a comparison of the perimeters of different ellipses, which, it surely seems to me, would overturn all of analysis.
My question: How might one find a comparison of perimeters of different ellipses if the differential equation were separable & why would this overturn all of analysis?
Links to the original latin & English translation of Euler's article: http://eulerarchive.maa.org/pages/E028.html
More context:
- I believe that the separation of variables in differential equations is so carefully sought because a solution of the equation follows directly from that discovery, which is evident to one practiced enough in these matters. Moreover the integration of differential equations, if indeed it succeeds, is begun best by separating variables. Though certainly innumerable equations have been given, whose integrals can be found without such a separation – the Celebrated Johann Bernoulli exhibited a method of this type in our Comm. Tom. I page 1672 – yet all of these equations have been arranged in such a way, so that either the separation of variables is obvious by itself, or that at least the separation may be derived from that integration. It is indeed likely that the computation of solutions that Analysts have found up until now are all of this type of equation, that, even if the variables can be separated in no other way, a separation still arises from that solution. For this reason, I have believed until now that no solvable differential equation could be produced whose separation would elude all men.
- Recently however while engaged in the rectifying of an ellipse, I unexpectedly came upon a differential equation by which I was able to solve the rectification of the ellipse, yet a separation of variables could not be found, not even from the method of solution. In fact the equation I obtained was $dy + \frac{y^2dx}{x} = \frac{x dx}{x^2−1}$ which closely resembles the Riccati equation, and as it happens it is as difficult to separate as $dy + y^2 dx = x^2 dx$. This case seemed exceedingly paradoxical to me at first; but after studying the solution more carefully I realized easily not only that a separation could not be deduced from it, but also, that if a separation were to succeed by another method, far greater absurdities would follow. One might find a comparison of the perimeters of different ellipses, which, it surely seems to me, would overturn all of analysis. This solution moreover is extremely easy, it is completed indeed by the elongation of infinite ellipses having one of two axes in common, and for this reason it must be substantially preferred to the usual way of solving quadratures.