- Picone identity
In

mathematics , in the field ofordinary differential equation s, the**Picone identity**, named afterMauro 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 ofdifferential equation s anddifference equation s.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 =

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