
Sturm separation theorem
Encyclopedia

Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of homogeneous
Homogeneous differential equation
The term homogeneous differential equation has several distinct meanings.One meaning is that a first-order ordinary differential equation is homogeneous if it has the formwhere F is a homogeneous function of degree zero; that is to say, that F = F.In a related, but distinct, usage, the term linear...
second order linear differential equation
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
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 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 theoremSturm-Picone comparison theorem
In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential...
.
Proof
The proof is by contradiction. Assume that v has no zeros in ]x0, x1[. Since u and v are linearly independent, v cannot vanish at either x0 or x1, so the quotient u / v is well-defined on the closed interval [x0, x1], and it is zero at x0 and x1. Hence, by Rolle's theoremRolle's theorem
In calculus, Rolle's theorem essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero.-Standard version of the theorem:If a real-valued function ƒ is continuous on a closed...
, there is a point ξ between x0 and x1 where

vanishes. Hence,

On the other hand, there can be only one zero between x0 and x1, 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 .
An alternative proof
Since







So at


and either




and since
















Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
there exists


By the same reasoning as in the first proof,

