Sturm separation theorem

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 ${\displaystyle \displaystyle u}$ 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

${\displaystyle u(x)v'(x)-u'(x)v(x)\neq 0.}$

So at ${\displaystyle \displaystyle x=x_{0}}$

${\displaystyle W(x_{0})=-u'\left(x_{0}\right)v\left(x_{0}\right)}$

and either ${\displaystyle u'\left(x_{0}\right)}$ and ${\displaystyle v\left(x_{0}\right)}$ are both positive or both negative. Without loss of generality, suppose that they are both positive. Now, at ${\displaystyle \displaystyle x=x_{1}}$

${\displaystyle W(x_{1})=-u'\left(x_{1}\right)v\left(x_{1}\right)}$

and since ${\displaystyle \displaystyle x=x_{0}}$ and ${\displaystyle \displaystyle x=x_{1}}$ are successive zeros of ${\displaystyle \displaystyle u(x)}$ it causes ${\displaystyle u'\left(x_{1}\right)<0}$. Thus, to keep ${\displaystyle \displaystyle W(x)<0}$ we must have ${\displaystyle v\left(x_{1}\right)<0}$. We see this by observing that if ${\displaystyle \displaystyle u'(x)>0{\mbox{ }}\forall {\mbox{ }}x\in \left(x_{0},x_{1}\right]}$ then ${\displaystyle \displaystyle u(x)}$ would be increasing (away from the ${\displaystyle \displaystyle x}$-axis), which would never lead to a zero at ${\displaystyle \displaystyle x=x_{1}}$. So for a zero to occur at ${\displaystyle \displaystyle x=x_{1}}$ at most ${\displaystyle u'\left(x_{1}\right)=0}$ (i.e., ${\displaystyle u'\left(x_{1}\right)\leq 0}$ and it turns out, by our result from the Wronskian that ${\displaystyle u'\left(x_{1}\right)\leq 0}$). So somewhere in the interval ${\displaystyle \left(x_{0},x_{1}\right)}$ the sign of ${\displaystyle \displaystyle v(x)}$ changed. By the Intermediate Value Theorem there exists ${\displaystyle x^{*}\in \left(x_{0},x_{1}\right)}$ such that ${\displaystyle v\left(x^{*}\right)=0}$.

On the other hand, there can be only one zero in ${\displaystyle \left(x_{0},x_{1}\right)}$, because otherwise ${\displaystyle v}$ would have two zeros and there would be no zeros of ${\displaystyle u}$ in between, and it was just proved that this is impossible.

• Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.