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 $(2n+1)^2(m^2+4m) + 16n^2 + 16n=0$.

If this is too vague, are there other examples of differential equations which exhibit a similar behaviour with respect to elementary solutions?

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.

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?

If this is too vague, are there other examples of differential equations which exhibit a similar behaviour with respect to elementary solutions?

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 $(2n+1)^2(m^2+4m) + 16n^2 + 16n=0$.

If this is too vague, are there other examples of differential equations which exhibit a similar behaviour with respect to elementary solutions?