  In this paper we study the existence of a first zero and the oscillatory
behavior of solutions of the ordinary differential equation $(vz')'+Avz = 0$,
where $A,v$ are functions arising from geometry. In particular, we introduce a
new technique to estimate the distance between two consecutive zeros. These
results are applied in the setting of complete Riemannian manifolds: in
particular, we prove index bounds for certain Schr\"odinger operators, and an
estimate of the growth of the spectral radius of the Laplacian outside compact
sets when the volume growth is faster than exponential. Applications to the
geometry of complete minimal hypersurfaces of Euclidean space, to minimal
surfaces and to the Yamabe problem are discussed.
