In mathematics

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, the Picone identity, named after Mauro Picone

Mauro Picone

Mauro Picone was an Italian mathematician. He is known for the Picone identity, for the Sturm-Picone comparison theorem and for being the founder of the Istituto Nazionale per le Applicazioni del Calcolo, presently named after him...

, is a classical result about 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...

 linear second order differential equations. It is useful in studying the oscillation of such equations and has been generalized to other type of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s and difference equations.

The Picone identity is used to prove the Sturm–Picone comparison theorem.

Picone identity

Suppose that u and v are solutions of the two homogeneous linear second order differential equations in self adjoint form
and
Then, for all x with v(x) ≠ 0, the following identity holds