Picone identity

In mathematics, in the field of ordinary differential equations, the Picone identity, named after Mauro Picone, is a classical result about homogeneous linear second order differential equations. It is useful in studying the oscillation of such equations and has been generalized to other type of differential equations and difference equations.

The Picone identity is used to prove the Sturm-Picone comparison theorem.

Picone identity

Suppose that "u" and "v" are solutions of the two homogeneous linear second order differential equations in self adjoint form:$(p\_1(x)\; u\text{'})\text{'}\; +\; q\_1(x)\; u\; =\; 0\; ,$and:$(p\_2(x)\; v\text{'})\text{'}\; +\; q\_2(x)\; v\; =\; 0.\; ,$Then, for all "x" with "v"("x") ≠ 0, the following identity holds:$left(frac\{u\}\{v\}(p\_1\; u\text{'}\; v\; -\; p\_2\; u\; v\text{'})\; ight)\text{'}\; =\; left(q\_2\; -\; q\_1\; ight)\; u^2\; +\; left(p\_1\; -\; p\_2\; ight)u\text{'}^2\; +\; p\_2left(u\text{'}-v\text{'}frac\{u\}\{v\}\; ight)^2.$

ee also

* Sturm-Picone comparison theorem

References

* cite journal
last = Picone
first = Mauro
authorlink = Mauro Picone
coauthors =
title = Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare del secondo ordine
journal = Ann. Scuola Norm. Sup. Pisa
volume = 11
issue =
pages = 1–141
publisher =
date = 1910
url =
doi =
id =
accessdate =