Optimal uniform estimates and rigorous asymptotics beyond all orders for a class of ODE's

Abstract

For first order differential equations of the form $y^{\prime }={\sum }_{p=0}^P{F}_p(x)y^p$ and second order homogeneous linear differential equations $y^{\prime \prime }+a(x)y^{\prime }+b(x)y=0$ with locally integrable coefficients having asymptotic (possibly divergent) power series when $|x|\to \infty$ on a ray $\arg (x)=$const, under some further assumptions, it is shown that, on the given ray, there is a one-to-one correspondence between true solutions and (complete) formal solutions. The correspondence is based on asymptotic inequalities which are required to be uniform in $x$ and optimal with respect to certain weights.