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
#$(p\_1(x)\; y^prime)^prime\; +\; q\_1(x)\; y\; =\; 0\; ,$
#$(p\_2(x)\; y^prime)^prime\; +\; q\_2(x)\; y\; =\; 0\; ,$be two homogeneous linear second order differential equations in self adjoint form with:and:$q\_1(x)\; le\; q\_2(x).,$

Let "u" be a non-trivial solution of (1) with successive roots at "z"₁ and "z"₂ and let "v" be a non-trivial solution of (2). Then one of the following properties holds.
*There exists an "x" in ["z"₁, "z"₂] such that "v"("x") = 0; or
*there exists a λ in R such that "v"("x") = λ "u"("x").