Sturm separation theorem

In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating.

Given a homogeneous second order linear differential equation and two continuous linear independent solutions u(x) and v(x) with x₀ and x₁ successive roots of u(x), then v(x) has exactly one root in the open interval ]x₀, x₁[. It is a special case of the Sturm-Picone comparison theorem.

Since ${\displaystyle \displaystyle u}$ and ${\displaystyle \displaystyle v}$ are linearly independent it follows that the Wronskian ${\displaystyle \displaystyle W[u,v]}$ must satisfy ${\displaystyle W[u,v](x)\equiv W(x)\neq 0}$ for all ${\displaystyle \displaystyle x}$ where the differential equation is defined, say ${\displaystyle \displaystyle I}$. Without loss of generality, suppose that ${\displaystyle W(x)<0{\mbox{ }}\forall {\mbox{ }}x\in I}$. Then

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