In the mathematical modelling of many problems in pure and applied sciences, such as physics, engineering, celestial mechanics, among others, forced and damped oscillators appear. Although the precision of the method of the T-functions sequence is high, the measurement of their coefficients in each case involves complex recurrences. The T-functions series method is converted into a multi-step method whose coefficients are determined using recurrence processes to avoid this inconvenience. These methods are convergent, and the T-functions sequence approach has the same properties. There are numerical examples already used by other authors, such as a stiff problem, a Duffing oscillator, and a problem with an equatorial satellite when the disturbance comes from J2 zonal harmonics.

**Author(s) Details**

**M.****Cort´es-Molina
**Dept. of Applied Mathematics, Escuela Polit´ecnica Superior, University of Alicante, Spain.

**F. Garc´ıa-Alonso
**Dept. of Applied Mathematics, Escuela Polit´ecnica Superior, University of Alicante, Spain.

**J. A. Reyes
**Dept. of Applied Mathematics, Escuela Polit´ecnica Superior, University of Alicante, Spain.

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