In the field of
ordinary differential equations, the Mingarelli identity
 is a theorem that provides criteria for the
non-oscillation of solutions of some
linear differential equations in the real domain. It extends the
Picone identity from two to three or more differential equations of the second order.
Consider the n solutions of the following (uncoupled) system of second order linear differential equations over the t–interval [a, b]:
- where .
Let denote the forward difference operator, i.e.
The second order difference operator is found by iterating the first order operator as in
with a similar definition for the higher iterates. Leaving out the independent variable t for convenience, and assuming the xi(t) ≠ 0 on (a, b], there holds the identity,
When n = 2 this equality reduces to the
The above identity leads quickly to the following comparison theorem for three linear differential equations,
 which extends the classical
Sturm–Picone comparison theorem.
Let pi, qi i = 1, 2, 3, be real-valued continuous functions on the interval [a, b] and let
be three homogeneous linear second order differential equations in
self-adjoint form, where
- pi(t) > 0 for each i and for all t in [a, b] , and
- the Ri are arbitrary real numbers.
Assume that for all t in [a, b] we have,
Then, if x1(t) > 0 on [a, b] and x2(b) = 0, then any solution x3(t) has at least one zero in [a, b].