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

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For first order differential equations of the form y'=∑ₚ₌₀ᵖ Fₚ(x)yᵖ and second order homogeneous linear differential equations y''+a(x)y'+b(x)y=0 with locally integrable coefficients having asymptotic (possibly divergent) power series when |x|→∞ on a ray (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.