Solve this equation numerically with the initial conditions

Question

See the advance of the perihelion of Mercury It is possible to ‘see' the advance of the perihelion of Mercury predicted by general relativity by direct numerical solution. Take Einstein's path equation

(d²v/dθ²⁾ + v = (1/1-e²) + ηv²,

where υ = au. Here a and e are the semi-major axis and eccentricity of the non-relativistic elliptic orbit and η = 3MG/ac² is a small dimensionless parameter.

For the orbit of Mercury, η = 2.3 × 10^-7 approximately. Solve this equation numerically with the initial conditions r = a(1 + e) and r^· = 0 when θ = 0; this makes θ = 0 an aphelion of the orbit.

To make the precession easy to see, use a fairly eccentric ellipse and take η to be about 0.005, which speeds up the precession by a factor of more than 104!