Picone identity

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.

Picone identity

Given two homogeneous linear second order differential equations in self adjoint form

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

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

with u a solution of (1) and v a non trivial solution of (2), then for all x with v(x) ≠ 0 the following identity holds

$({\frac {u}{v}}(p_{1}u'v-p_{2}uv'))'=(q_{2}-q_{1})u^{2}+(p_{1}-p_{2})u'^{2}+p_{2}(u'-v'{\frac {u}{v}})^{2}$.

References

• Picone, Mauro (1910). "Sui valori eccezionalle di un parametro da cui dipende un' equatione differenzialle lineare del secondo ordine". Ann. Scuola Norm. Sup. Pisa. 11: 1–141. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help)