Sturm–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 certain linear differential equations.

Sturm-Picone comparison theorem

Let

(1)$(p_{1}(x)y')'+q_{1}(x)y=0$

(2)$(p_{2}(x)y')'+q_{2}(x)y=0$

be two homogeneous linear second order differential equations in self adjoint form with

$0<p_{2}(x)\leq p_{1}(x)$

and

$q_{1}(x)\leq q_{2}(x).$

Let u be a non trivial solution of (1) with successive roots at z₁ and z₂ and v be a non trivial solution of (2) then one of the following properties holds

• there exists an x in ]z₁,z₂[ so that v(x) = 0
• there exists a λ in R so that v(x) = λ u(x)