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| In the field of [[ordinary differential equation]]s, the '''Picone identity''', named after [[Mauro Picone]],<ref name="Picone">{{harvnb|Picone|1910}}</ref> is a classical result about [[homogeneous differential equation|homogeneous]] linear second order differential equations. Since its inception in 1910 it has been used with tremendous success in association with an almost immediate proof of the [[Sturm comparison theorem]], a theorem whose proof took up many pages in Sturm's original memoir of 1836. It is also useful in studying the [[oscillation (differential equation)|oscillation]] of such equations and has been generalized to other type of [[differential equation]]s and [[difference equation]]s.
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| The Picone identity is used to prove the [[Sturm comparison theorem]] (see also [[Sturm–Picone comparison theorem]]).
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| == Picone identity ==
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| Suppose that ''u'' and ''v'' are solutions of the two homogeneous linear second order differential equations in [[self-adjoint form]]
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| :<math>(p_1(x) u')' + q_1(x) u = 0 \, </math>
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| and
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| :<math>(p_2(x) v')' + q_2(x) v = 0. \,</math> | |
| Then, for all ''x'' with ''v''(''x'') ≠ 0, the following identity holds
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| :<math>\left(\frac{u}{v}(p_1 u' v - p_2 u v')\right)' = \left(q_2 - q_1\right) u^2 + \left(p_1 - p_2\right)u'^2 + p_2\left(u'-v'\frac{u}{v}\right)^2.</math>
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| == References ==
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| {{Reflist}}
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| * {{cite journal
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| | last = Picone
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| | first = Mauro
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| | authorlink = Mauro Picone
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| | title = Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare del secondo ordine
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| | journal = Ann. Scuola Norm. Sup. Pisa
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| | volume = 11
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| | pages = 1–141
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| | year = 1910
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| | ref=harv}}
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| * {{cite journal
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| | last = Swanson
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| | first = Charles A.
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| | title = Picone's Identity
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| | journal = Rendiconti di Matematica
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| | volume = 8
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| | issue = 2
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| | pages = 373–397
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| | year = 1975
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| | ref=harv}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Mathematical identities]]
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