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In Gian-Carlo Rota's article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations", at the end of the third lesson he states a theorem:

"Every differential polynomial in the two solutions of a second order linear differential equation which is independent of the choice of a basis of solutions equals a polynomial in the Wronskian and in the coefficients of the differential equation."

Does anyone have a reference (with proof) for this theorem?

Thank you.

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  • $\begingroup$ This is indeed what is stated, say here: web.williams.edu/Mathematics/lg5/Rota.pdf But what does that even mean? Are we considering $p(u,v)$ such that whatever basis $u,v$ of the solution space I come up with, $p(u,v)$ comes out the same? (Surely not.) $\endgroup$
    – Christian Remling
    Commented Sep 15 at 15:32
  • $\begingroup$ Yes, I am similarly confused, part of why I am asking. $\endgroup$
    – Ryan
    Commented Sep 15 at 17:02

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