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I am currently trying to understand Euler's article E71 on the Riccati differential equation and its connection with continued fractions. Apparently Daniel Bernoulli had shown that the equation $$ y' + ay^2 = bx^\alpha $$ can be solved by elementary functions if $\alpha = -2$ or $\alpha = - \frac{4n}{2n-1}$ for integers $n$. The proof proceeds by reduction: using a series of transformations (which you can find e.g. in Kamke's classical book on differential equations), the differential equation is transformed into $$ y' + Ay^2 = Bx^\beta $$ with $$ \beta = \frac{\alpha+4}{\alpha+3} = \frac{4(n-1)}{2(n-1)-1}. $$ Bernoulli claimed and Liouville proved that the equation cannot be solved by elementary functions for other exponents except $\alpha = -2$, the limit point for $n \to \infty$.

To a number theorist, this looks a little bit like descent: every exponent $\alpha$ for which there is an elementary solution can be reached from $\alpha = 0$ by a finite series of transformations.

What I would like to know is:

Is this a superficial resemblance, or is there something deeper going on?

In particular, can we attach some algebraic curve to this differential equation in such a way that its rational points correspond to the exponents for which the Riccati equation has an elementary solution, and can the transformations be explained geometrically? For what it's worth, in Liouville's article we can find the singular quartic $(2n+1)^2(m^2+4m) + 16n^2 + 16n=0$.

Edit. I have just discovered an article on the Riccati equation from a group theoretical viewpoint which seems to confirm my suspicion that there is a lot of algebra beneath the integrability of the Riccati equation.

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Franz, can you, please, post a link to E71 (if it exists) or explain the nomenclature? –  Victor Protsak Sep 30 '10 at 16:42
    
Done. E71 is Euler's manuscript no. 71 in the Enestrom index. –  Franz Lemmermeyer Sep 30 '10 at 16:59
    
I'd settle for a reference where the explicit solutions are given. –  Deane Yang Sep 30 '10 at 19:20
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Hi Franz, I'm studying something similar as well, and since I've only begun digesting the concepts I can't give an answer; the best I can do now is to point you to the book "Continued Fractions with Applications" by Lorentzen and Waadeland. The CF solution of a Riccati DE is discussed starting on page 540 of the book. Hope this helps. –  J. M. Oct 1 '10 at 0:03

2 Answers 2

(NOT AN ANSWER)

I can't answer this question at all, but I wanted to note that the Riccati equation is important to differential geometers, because it is the equation satisfied by the second fundamental form of level sets of the distance function along a geodesic. My immediate instinct is tosubstitute $y = u'/(au)$ (here, $y$ is the second fundamental form, and $u$ is the Jacobi field) to get a linear second order equation. Here, that would be $u'' = abx^\alpha u$. The function $ab x^\alpha$ is the sectional curvature on the $2$-plane spanned by the tangent to the geodesic and the Jacobi field. The $\alpha = -2$ case is familiar, because it corresponds to the most common type of point singularity (by setting $y = cr^p$ for any $p \ne 1$). The other cases seem like they might also be interesting to differential geometers.

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this is not an answer, i am not sure if i can add this as a comment to your question. so i typed it here.

from treatise on differential equation by a. r. forsyth in section 109 it is shown that $y^\prime + a y^2 = b x^\alpha$ is integrable if $\alpha = -\frac{4k}{2k \pm 1}$ for $k$ a positive integer or zero.

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