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[[File:Airy plot.svg|thumb|250px|The zeros of two linearly independent solutions of the [[Airy equation]] <math>y'' - xy = 0</math> alternate, as predicted by the Sturm separation theorem.]]
In [[mathematics]], in the field of [[ordinary differential equation]]s, '''Sturm separation theorem''', named after [[Jacques Charles François Sturm]], describes the location of roots of [[homogeneous differential equation|homogeneous]] [[order (differential equation)|second order]] [[linear differential equation]]s. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating.

== Sturm separation theorem ==

Given a homogeneous second order linear differential equation and two continuous linear independent solutions ''u''(''x'') and ''v''(''x'') with ''x''<sub>0</sub> and ''x''<sub>1</sub> successive roots of ''u''(''x''), then ''v''(''x'') has exactly one root in the open interval ]''x''<sub>0</sub>, ''x''<sub>1</sub>[. It is a special case of the [[Sturm-Picone comparison theorem]].

== Proof ==

Since <math>\displaystyle u</math> and <math>\displaystyle v</math> are linearly independent it follows that the Wronskian <math>\displaystyle W[u,v]</math> must satisfy <math>W[u,v](x)\equiv W(x)\neq 0</math> for all <math>\displaystyle x</math> where the differential equation is defined, say <math>\displaystyle I</math>. Without loss of generality, suppose that <math>W(x)<0\mbox{ }\forall\mbox{ }x\in I</math>. Then
:<math>u(x)v'(x)-u'(x)v(x)\neq 0.</math>
So at <math>\displaystyle x=x_0</math>
:<math>W(x_0)=-u'\left(x_0\right)v\left(x_0\right)</math>
and either <math>u'\left(x_0\right)</math> and <math>v\left(x_0\right)</math> are both positive or both negative. Without loss of generality, suppose that they are both positive. Now, at <math>\displaystyle x=x_1</math> 
:<math>W(x_1)=-u'\left(x_1\right)v\left(x_1\right)</math>
and since <math>\displaystyle x=x_0</math> and <math>\displaystyle x=x_1</math> are successive zeros of <math>\displaystyle u(x)</math> it causes <math>u'\left(x_1\right)<0</math>. Thus, to keep <math>\displaystyle W(x)<0</math> we must have <math>v\left(x_1\right)<0</math>. We see this by observing that if <math>\displaystyle u'(x)>0\mbox{ }\forall\mbox{ }x\in \left(x_0,x_1\right]</math> then <math>\displaystyle u(x)</math> would be increasing (away from the <math>\displaystyle x</math>-axis), which would never lead to a zero at <math>\displaystyle x=x_1</math>. So for a zero to occur at <math>\displaystyle x=x_1</math> at most <math>u'\left(x_1\right)=0</math> (i.e., <math>u'\left(x_1\right)\leq 0</math> and it turns out, by our result from the Wronskian that <math>u'\left(x_1\right)\leq 0</math>). So somewhere in the interval <math>\left(x_0,x_1\right)</math> the sign of <math>\displaystyle v(x)</math> changed. By the [[Intermediate Value Theorem]] there exists <math>x^*\in\left(x_0,x_1\right)</math> such that <math>v\left(x^*\right)=0</math>.

On the other hand, there can be only one zero in <math>\left(x_0,x_1\right)</math>, because otherwise v would have two zeros and there would be no zeros of u in between, and it was just proved that this is impossible.

== References ==

*{{cite book| last = Teschl| given = G.|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

[[Category:Ordinary differential equations]]
[[Category:Theorems in analysis]]