  We build an existence theory for nonoscillatory second order differential
equations of the form (A) $(p(t)x')' = q(t)x, $ $p(t)$ and $q(t)$ being
positive continuous functions on $[a,\infty)$, in which a crucial role is
played by a pair of the Riccati differential equations (R1) $u' = q(t) -
u^2/p(t)$, (R2) $ v' = 1/p(t) - q(t)v^2$, associated with (A). An essential
part of the theory is the construction of a pair of linearly independent
nonoscillatory solutions $x_1(t)$ and $x_2(t)$ of (A) enjoying explicit
exponential-integral representations in terms of solutions $u_1(t)$ and
$u_2(t)$ of (R1) or in terms of solutions $v_1(t)$ and $v_2(t)$ of (R2).
