Heuristic asymptotic methods are commonly used to study the long-term development of oscillatory solutions of nonlinear ordinary differential equations and wave-type solutions of partial differential equation. For certain classes of weakly nonlinear systems, energy methods are here used to establish the validity of such approximations. There is an overall limitation of the results to a time interval determined by the time scale of significant energy transfer, but this is sufficiently long for interesting physical effects to be discussed. The basic results take the form that, when the heuristic methods yield an approximation giving a uniformly bounded small error in the differential equation, then the error in the solution is small. They do not depend strongly on the properties of the approximation, other than on simple bounds. The possibility of extending these methods to more complicated physical systems is considered. From these considerations some doubts are raised about the validity of averaging methods when applied to strongly nonlinear systems. Even for weakly nonlinear systems there are possibilities of error arising from sources that do not appear to have been considered previously. An example is given where the averaged solution is grossly in error.