An inequality for the amplitudes and areas in vibration diagrams of time-dependent frequency
but (1) cannot be integrated in explicit terms in this case either). It is therefore of interest that, without solving (1), one can delimit an explicit inequality for the maximum amplitude and the area of every half-wave (if any) of an arbitrary solution of every differential equation of the form (1). This is the content of (4) below. It will also be shown that the inequality in question is the best possible of its kind (if w(<) is unspecified). Unless the given ^-interval is either too short or such that co(f) is "too close to 0" on it (i.e., (1) is "too close to x" = 0," with x(t) = cxt + c2), a solution x(t) will have more than one zero, say x(a) = 0 and x(b) = 0, where a < b. Then, since x(t) is not the trivial solution, it can be assumed that t = b is the first zero following the zero t = a; so that x(t) 0 if a < t < b. Since —x(t) is a solution of (1) if x{t) is, there is no further loss of generality in assuming that