  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.

Classical Analysis and ODEs