In the field of

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

s, the Picone identity, named after

Mauro Picone Mauro Picone (2 May 1885 – 11 April 1977) was an Italian mathematician. He is known for the Picone identity, the Sturm-Picone comparison theorem and being the founder of the Istituto per le Applicazioni del Calcolo, presently named after hi ...

, is a classical result about

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

linear second order differential equations. Since its inception in 1910 it has been used with tremendous success in association with an almost immediate proof of the

Sturm comparison theorem Sturm (German for storm) may refer to: People * Sturm (surname), surname (includes a list) * Saint Sturm (died 779), 8th-century monk Food * Federweisser, known as ''Sturm'' in Austria, wine in the fermentation stage * Sturm Foods, an American d ...

, a theorem whose proof took up many pages in Sturm's original memoir of 1836. It is also useful in studying the

oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

of such equations and has been generalized to other type of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s and

difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s. 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 In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...

(p_1(x) u')' + q_1(x) u = 0

and :

(p_2(x) v')' + q_2(x) v = 0.

Then, for all ''x'' with ''v''(''x'') ≠ 0, the following identity holds :

\left(\frac(p_1 u' v - p_2 u v')\right)' = \left(q_2 - q_1\right) u^2 + \left(p_1 - p_2\right)u'^2 + p_2\left(u'-v'\frac\right)^2.

Proof

\left(\frac(p_1 u' v - p_2 u v')\right)'=\left(u p_1 u' -p_2v'u^2 \frac 1 v \right)'
=u'p_1u' +u(p_1 u')' -(p_2v')'u^2 \frac 1 v-p_2 v' 2u u' \frac 1 v +p_2 v' u^2 \frac=

= p_1u'^2-2p_2\frac v+p_2\frac+u (p_1u')'- (p_2 v')'\frac=

= p_1u'^2-p_2u'^2+p_2u'^2-2p_2u'\frac  v+p_2\left(\frac\right)^2-u (q_1u)+ (q_2 v)\frac  v 
= \left(p_1 - p_2\right)u'^2 + p_2\left(u'-v'\frac\right)^2 + \left(q_2 - q_1\right) u^2

Notes

* *

References

{{Reflist Ordinary differential equations Mathematical identities