In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation is called oscillating if it has an infinite number of roots; otherwise it is called non-oscillating.
The number of roots carries also information on the spectrum of associated boundary value problems.
Oscillation theory was initiated by Jacques Charles François Sturm in his investigations of Sturm–Liouville problems from 1836.
There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots.
The investigation of the number of roots of the Wronski determinant of two solutions is known as relative oscillation theory.