As x i (t 0) and ⁠ d x i (t 0) / dt ⁠ are given as initial conditions, every ⁠ d x i (t) / dt ⁠ is known. Differentiating ⁠ d x i (t) / dt ⁠ results in ⁠ d x i (t) / dt ⁠ which at t 0 which is also known, and the Taylor series is constructed iteratively.