FAMILIES OF COMBINATORY SOLUTIONS OF Y' = 1-EQUATIONS(Y2 AND AUTONOMOUS DIFFERENTIAL)

Let A = {X,E2,E3,C3,...} be the set of all atomic species (up to isomo rphism). The differential ring Q[[A]] of all rational species, equippe d with the substitution operation, is a natural extension of the ring Q[[X]] of formal power series in one variable. We present a general me thod for finding, in Q[[A]], the infinite family of solutions of the a utonomous equation Y' = F(Y), Y(0) = 0 (F is-an-element-of Q[[A]], F(0 ) not-equal 0). We obtain, in particular, an infinite number of liftin gs of Y = tan X which is the solution, in Q[[X]], of Y' = 1 + Y2, Y(0) = 0. We also show that this last equation has no solution in N[[A]]. Our study completes the work done in the context of L-species, where L eroux and Viennot have shown that such equations have a unique solutio n.