3469 - Assignment 2 solution Mike Peardon (mjp@maths.tcd.ie) Michaelmas Term 2015-16 Question 1 — Shooting method There are 11 solutions to the second-order ODE given by d2 x 2 = −x3 te−x /2 with x(0) = 0, x(10) = 1. 2 dt (1) The values of ẋ(0) that give these solutions are specified in Table 1 and all 11 solutions are plotted in Fig. 1. 5 4 3 2 x(t) 1 0 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 t Figure 1: The 11 solutions to the ODE specified in Eqn. 1 Table 1: The 11 values for ẋ(0) for the solutions to Eqn. 1, k = 1, 2, . . . 11 (checked to 7sf) k 1 2 3 4 5 6 ẋk (0) k -2.037797 7 -2.033002 8 -2.017598 9 -1.996646 10 -1.829865 11 -1.265691 1 ẋk (0) 1.919303 1.981126 2.022206 2.031362 2.038403 Question 2 — Planets The trajectories of the four planets are shown in Fig. 2 For 3sf precision, about 10,000 leap-frog 1 Planet 0 Planet 1 Planet 2 Planet 3 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 Figure 2: The trajectories of the four planets Table 2: The locations of the four planets at t = 3 (to 3sf) Planet 0 1 2 3 Position x0 x1 -0.309 -0.353 -0.318 -0.525 0.0360 0.907 0.102 0.859 steps are needed. 2