Quasianalytic solutions of differential equations at singular points

We consider differential equations of the form F(x, y(x), y '(x)) = 0. Here x denotes a real variable and x 7 -> y(x) a real m-vector function. Furthermore, F is a C infinity m-vector function defined in a neighbourhood of the origin in R x Rm x Rm such that F (0, 0, 0) = 0. We also suppose that F is in some Denjoy-Carleman quasianalytic class. If the equation F(x, y(x), y '(x)) = 0 has a formal solution u(x) = (u1(x), ... , um(x)), where each uj(x) = sigma infinity p=2 aj,pxp is an element of R[[x]], j = 1, ... , m, is a formal power series with real coefficients, we give a condition that guarantees that each uj(x) is the Taylor expansion of a function in the same Denjoy-Carleman quasianalytic class as F. By quasianalyticity, we obtain a solution of the differential equation F(x, y, y ') = 0 which is in the same Denjoy- Carleman quasianalytic class as F. Unfortunately, this condition is rather restrictive as regards the behaviour of the solutions in a neighbourhood of the origin in R. It can be seen from simple examples that the condition cannot be relaxed.