Generalized finite moments and Liouville–Green approximations

The Liouville–Green approximation for the differential equation z ̈ ( t ) + f ( t ) z ( t ) = 0 is exhaustively studied. New moments of f –and not only the first and second–are proposed, allowing asymptotic formulae with explicit estimates for the error functions. The recessive and Dominant character of the fundamental system of solutions is clear. Our study includes the pointwise condition lim t → ∞ t 2 f ( t ) = a . Assuming, in this case, that t 2 f ( t ) has bounded variation, we prove natural Liouville–Green approximations which establish the oscillatory character of the solutions for a > 1 / 4 and the non-oscillatory one for a < 1 / 4 . The method used seems new and it is based on the analysis of an integral equation for the error function and of a parametric Riccati expression.