Remarks on the uniqueness of second order ODEs

We are concerned with the uniqueness problem for solutions to the second order ODE of the form x″+f(x, t) = 0, subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x, for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable.