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Let be the ring of power series over an effective ring . Brent has first shown that differential equations over may be solved in an asymptotically efficient way using Newton's method. More precisely, if denotes the complexity in order two polynomials of degree over , then the first coefficients of the solution can be computed in time . However, this complexity does not take into account the dependency of on the order of the equation, which is exponential for the original method and linear for a recent improvement. In this paper, we present a technique to get rid of this constant factor, by applying Newton's method up to an order like and trading the remaining Newton steps against a lazy or relaxed algorithm in a suitable FFT model.

**Keywords: **power series, Newton's method, differential
equation, FFT

**A.M.S. subject classification: **68W25, 37M99, 90C53,
42-04, 68W30, 33F05