What theorem of Liouville's is Gian-Carlo Rota referring to here?

Question

I am very curious about this remark in Lesson Four of Rota's talk, Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations:

"For second order linear differential equations, formulas for changes of dependent and independent variables are known, but such formulas are not to be found in any book written in this century, even though they are of the utmost usefulness.

"Liouville discovered a differential polynomial in the coefficients of a second order linear differential equation which he called the invariant. He proved that two linear second order differential equations can be transformed into each other by changes of variables if and only if they have the same invariant. This theorem is not to be found in any text. It was stated as an exercise in the first edition of my book, but my coauthor insisted that it be omitted from later editions."

Does anyone know where to find this theorem?

Answer 1

See E. Hille, Ordinary differential equations in the complex domain, Wiley, New York, 1976. The Liouville transformation is given on Page 179. The invariant mentioned by Rota is the function $Q(z)$ appearing as a coefficient of the equation in the canonical form.

Answer 2

This question had been bothering me for a while since I teach the intro differential equations courses occasionally, so I finally looked up the reference Anatoly gave and figured out the details. I'll drop them here in case anyone else can get use out of them.

Starting with the second-order equation

$y'' + Py' + Qy = 0$

we can make the change of variables $w = y \cdot e^{\int P/2}$. This change of variables is pretty clever; if you work it out, it happens to eliminate the first derivative term and gives us a new second-order equation of the form

$w'' + Q_0w = 0$.

If you calculate it out, you can find that $Q_0 = Q - \frac14P^2 - \frac12P'$. Nothing fancy here, just what happens when you do the change of variables. This $Q_0$ is the invariant of the second-order equation that is mentioned in the question.

Any two second-order equations that have the same invariant can easily be transformed into one another by a change of variables; simply change variables once to get to the standard form $w'' + Q_0w = 0$ and then change variables back into the other one.

More difficult is that all changes of variables preserve this invariant; proving it for changes of variables y = G x is an easy computation with a bunch of cancellation, but I'm not sure if we need to do more than that to finish the proof.

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From the point of view of someone teaching introductory differential equations, you are normally dealing with second-order equations $y'' + Py' + Qy = 0$ where $P$ and $Q$ are real numbers. In this case, the invariant $Q_0$ is just $-\frac14$ times the discriminant of the auxiliary equation. So the theorem says that any two equations with the same auxiliary equation discriminant can be transformed into each other.

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For example,

$y'' + 6y' + 10y = 0$

has $Q_0 \equiv 1$, so

it must be able to turn into

$w'' + 1w = 0$

via a change of variables. Indeed, if you let

$w = y e^{3x}$

then you get the equation $w'' + w = 0$, and the solution is

$y \cdot e^{3x} = c_1 \cos x + c_2 \sin x$

$y = c_1 e^{-3x} \cos x + c_2 e^{-3x}\sin x$.

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Another example:

$y'' + 6y' + 9y = 0$

has $Q_0 \equiv 0$,

so it must be able to turn into simply

$w'' = 0$

via a change of variables. The change of variables only depends on $P$, and yes, $w = y e^{3x}$ is a pretty good change of variables. Via this route we end up with

$y \cdot e^{3x} = c_1 + c_2 x$

$y = c_1 e^{-3x} + c_2 x e^{-3x}$

which is of course correct.

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So, in our intro differential equations classes, the invariant is just the familiar fact that if we complete the square of the auxiliary equation, we can see the correct change of variables that will leave us with a bunch of $\cos$, $\sin$, $\cosh$, and $\sinh$ in addition to our exponentials from $P/2$.

Answer 3

Kamke's classic compendium [1] of ODE solutions and solution methods displays this invariant in Part I, equation §25.1(4). The invariant is given for the more famous equations (Bessel, Legendre, hypergeometric, ...) of the large list of second order linear equations in Part III Chapter II.

[1] Kamke, E. Differentialgleichungen: Lösungen und Lösungernethoden. Vol. 1. First published in 1944.

Unfortunately, it seems that this book never appeared in English translation.