Smoothness of solutions of a nonlinear ode

Smoothness of a C(infinity)–function f is measured by (Carleman) sequence {M k} 0 (infinity); we say f (element of) C M (infinity) [0, 1] if |f(k)(t)| <= CR(sup k) M(k), k=0, 1, ... with C, R > 0. A typical statement proven in this paper is THEOREM: Let u, b be two C(infinity)-functions on [0, 1] such that (a) uʹ=u^(2)+b, (b) |b^(k)(t)| <= CR^k (k!)^(gamma), (gamma)>1, k (element of) Z+. Then |u^(k)(t)| <= C1R^k((k–1)!)^(gamma),k(element of) Z+.