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](http://www.ega-math.narod.ru/Tasks/GCRota.htm):



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> "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.
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> "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."
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Does anyone know where to find this theorem?



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


