Picone identity

From Wikipedia, the free encyclopedia

In mathematics, in the field of ordinary differential equations, the Picone identity, named after Mauro Picone, is a classical result about homogeneous linear second order differential equations. It is useful in studying the oscillation of such equations and has been generalized to other type of differential equations and difference equations.

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

[edit] Picone identity

Suppose that u and v are solutions of the two homogeneous linear second order differential equations in self adjoint form

(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{u}{v}(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{u}{v}\right)^2.

[edit] See also

[edit] References

  • Picone, Mauro (1910). "Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare del secondo ordine". Ann. Scuola Norm. Sup. Pisa 11: 1–141.