A priori analysis of initial data for the Riccati equation and asymptotic properties of its solutions

We obtain two main results for the Cauchy problem y'(x)+1/r(x)y(2) = q(x), y(x)vertical bar(x=x0)=yo, where x(0), y(0) is an element of R, r > 0, q >= 0, 1/r is an element of L-1(loc)(R), q is an element of L-1(loc)(R) and integral(x)(-infinity) 1/r(t) integral(x)(t) q(xi)d xi dt = integral(infinity)(x) 1/r(t) integral(t)(x) q(xi) d xi dt = infinity for all x is an element of R (1) For given initial data x(0), y(0) and functions r and q, we give a condition that can be used to determine whether the solution of the problem can be continued to the whole of R. (2) When the solution is defined on an infinite interval, we study its asymptotic properties as the argument tends to infinity.