Auxiliary fraction - Revision history

en>Magioladitis: ++

2015-01-06T09:39:47Z

← Older revision		Revision as of 11:39, 6 January 2015
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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~~.		Hi there. Let me begin by introducing the author, her name is Sophia Boon but she never truly liked that name. I am an invoicing officer and I'll be promoted quickly. Some time in the past he selected to reside in North Carolina and he doesn't plan on changing it. To perform lacross is some thing I really appreciate performing.<br><br>Here is my web blog; [http://conniecolin.com/xe/community/24580 online psychic chat]

	~~The Picone identity~~ is ~~used to prove the [[Sturm comparison theorem]] (see also [[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]]~~
	:<~~math~~>~~(p_1(x) u')' + q_1(x) u = 0 \,~~ <~~/math~~>
	~~and~~
	:~~<math>(p_2(x) v')' + q_2(x) v = 0~~. ~~\,<~~/~~math>~~
	~~Then, for all ''x'' with ''v''(''x'') ≠ 0, the following identity holds~~
	~~:<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>~~

	~~== References ==~~
	~~{{Reflist}}~~
	* {{cite journal
	~~\| last = Picone~~
	~~\| first = Mauro~~
	~~\| authorlink = Mauro Picone~~
	~~\| 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~~
	~~\| pages = 1–141~~
	~~\| year = 1910~~
	~~\| ref=harv}}~~
	* {{cite journal
	~~\| last = Swanson~~
	~~\| first = Charles A.~~
	~~\| title = Picone's Identity~~
	~~\| journal = Rendiconti di Matematica~~
	~~\| volume = 8~~
	~~\| issue = 2~~
	~~\| pages = 373–397~~
	~~\| year = 1975~~
	~~\| ref=harv}}~~

	~~[[Category:Ordinary differential equations]]~~
	~~[[Category:Mathematical identities]~~]

en>Bejnar: add ref

2014-01-07T22:42:07Z

add ref

← Older revision		Revision as of 00:42, 8 January 2014
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	~~The writer~~'s name is ~~Andera and she thinks~~ it ~~sounds quite good. Office supervising is exactly where her primary earnings comes from but she~~'s ~~currently applied for another 1~~. ~~I am truly fond~~ of ~~handwriting but I can't make it my profession truly~~. ~~North Carolina~~ is ~~where we~~'~~ve been~~ [~~http~~:/~~/www~~.~~octionx.sinfauganda.co.ug~~/~~node/22469 clairvoyant psychic] residing~~ for ~~years and will never transfer.~~<br><br>~~Feel free to visit my web blog: [http://www~~.~~chk~~.~~woobi~~.co.~~kr/xe/?document_srl~~=~~346069 cheap psychic readings~~] ~~readers;~~ [~~http~~:~~//fashionlinked.com/index.php?do=/profile-13453/info/ Read Much more~~],		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.

			The Picone identity is used to prove the [[Sturm comparison theorem]] (see also [[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]]
			:<math>(p_1(x) u')' + q_1(x) u = 0 \, </math>
			and
			:<math>(p_2(x) v')' + q_2(x) v = 0. \,</math>
			Then, for all ''x'' with ''v''(''x'') ≠ 0, the following identity holds
			:<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>

			== References ==
			{{Reflist}}
			* {{cite journal
			\| last = Picone
			\| first = Mauro
			\| authorlink = Mauro Picone
			\| 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
			\| pages = 1–141
			\| year = 1910
			\| ref=harv}}
			* {{cite journal
			\| last = Swanson
			\| first = Charles A.
			\| title = Picone's Identity
			\| journal = Rendiconti di Matematica
			\| volume = 8
			\| issue = 2
			\| pages = 373–397
			\| year = 1975
			\| ref=harv}}

			[[Category:Ordinary differential equations]]
			[[Category:Mathematical identities]]

en>Michael Hardy: /* Generating a Denominator with More Terminal Nines */ fixed these glaringly incorrect capitals; see WP:MOS

2012-07-15T19:00:14Z

Generating a Denominator with More Terminal Nines: fixed these glaringly incorrect capitals; see WP:MOS

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