A criterion for stable characteristic exponents
where g(t) and h(t) have the period, 1, oif {t) and do not vanish identically. The numbers + X occurring in (2), the characteristic exponents of (1), are determined mod 2nri only. If X is not a multiple of iri, it is clear that the two solutions (2) are linearly independent. If X is a multiple of iri, it depends on the elementary divisors of certain linear substitutions whether (1) does or does not possess two linearly independent solutions of the form (2) (in the second case, the general solution of (1) contains a secular term, that is, a term containing the factor t). Let the differential equation (1) be called of stable type if every solution, x(t), remains bounded as t—*± °°, and let the characteristic exponents +X of (1) be called of stable type if they are purely imaginary (including 0). Then it is clear that (1) cannot be of stable type unless the characteristic exponents are, and that (1) must be of stable type if the characteristic exponents are of stable type and distinct. Needless to say, the two characteristic exponents, X and —X, are considered as distinct if and only if they are distinct mod 2iri (that is, if X is not a multiple of iri). 2. The usual way of calculating X is supplied by Hill's method of infinite determinants (the zeros of the resulting transcendental equation being all possible values,