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

For first order differential equations of the form $y'=\sum_{p=0}^P F_p(x)y^p$ 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 $|x|\to\infty$ 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.