We determine conditions on the coefficients of both second and fourth order differential operators which give disconjugacy. The conditions exploit the change of sign of the coefficients to show how the change of sign enhances disconjugacy. An application is given to the stability of solutions of equations with periodic coefficients. Focal point and other boundary conditions are considered as well, and sufficient conditions are given which imply the nonexistence of nontrivial solutions of the differential equations satisfying the boundary conditions. Some open problems are stated for the minimization of certain nonlinear functionals.