7.3.9 (Series approximation for a closed orbit) In Example 7.3.1, we used the Poincaré-Bendixson Theorem to prove that the system ŕ=r(1-r²)+ μr cose, *=1 has a closed orbit in the annulus √√1-μ<r<√1+μ for all μ < 1. a) To approximate the shape r(e) of the orbit for μ << 1, assume a power series solution of the form r(0) = 1 + μr, (0) + O(²). Substitute the series into a differential equation for dr/de. Neglect all O(2) terms, and thereby derive a simple differential equation for r,(e). Solve this equation explicitly for r, (e). (The approximation technique used here is called regular perturbation theory; see Section 7.6.) b) Find the maximum and minimum r on your approximate orbit, and hence show that it lies in the annulus √1-μ<r<√1+μ, as expected. c) Use a computer to calculate r(0) numerically for various small μ, and plot the results on the same graph as your analytical approximation for r(e). How does the maximum error depend on μ?