2. *The perturbed Kepler*

This problem describes the motion of a planet according to Einstein's general relativity theory and the Schwarzschild potential is applied. The equations are:

$$\begin{aligned} \,^1y'' &= -\frac{\,^1y}{\sqrt{\,^1y^2 + \,^2y^2}^3} - (2+\delta)\delta \frac{\,^1y}{\sqrt{\,^1y^2 + \,^2y^2}^5},\\ \,^2y'' &= -\frac{\,^2y}{\sqrt{\,^1y^2 + \,^2y^2}^3} - (2+\delta)\delta \frac{\,^2y}{\sqrt{\,^1y^2 + \,^2y^2}^5},\end{aligned}$$

and the analytical solution is

$$y^1y = \cos(\mathbf{x} + \delta\mathbf{x}), \;^2y = \sin(\mathbf{x} + \delta\mathbf{x}).$$

We transformed this problem into a system of four first-order equations and solved for *xend* = 10*π* and *xend* = 20*π*. After recording the endpoint errors and the costs, we present the efficiency measures ratios of DLMP6(5) vs. NEW6(5) for perturbed Kepler in Tables 5 and 6.


**Table 5.** Efficiency measures ratios of DLMP6(5) vs. NEW6(5) for perturbed Kepler in [0, 10*π*].


**Table 6.** Efficiency measures ratios of DLMP6(5) vs. NEW6(5) for perturbed Kepler in [0, 20*π*].
