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 solutions 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.

If u(x) and v(x) are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with x0 and x1 being 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}

are linearly independent it follows that the Wronskian

must satisfy

[ u , v ] ( x ) ≡

{\displaystyle W[u,v](x)\equiv W(x)\neq 0}

for all

x

where the differential equation is defined, say

Without loss of generality, suppose that

x =

are both positive or both negative.

Without loss of generality, suppose that they are both positive.

are successive zeros of

We see this by observing that if

would be increasing (away from the

-axis), which would never lead to a zero at

So for a zero to occur at

and it turns out, by our result from the Wronskian that

So somewhere in the interval

the sign of

changed.

By the Intermediate Value Theorem there exists

On the other hand, there can be only one zero in

would have two zeros and there would be no zeros of

in between, and it was just proved that this is impossible.