Exercise 1.
Graph several solutions to dx/dt = y, dy/dt = -k^{2} sin x - γy, with k = 1, including at least the three solutions with the initial conditions (x(0),y(0)) = (0,1.5), (0,2), and (0,2.5), for0 ≤ t ≤ 8, the following ways:
a. In the x,y plane (i.e., the phase plane).
b. As x and y vs. t, with all graphs on one three dimensional coordinate system.
c. As x vs. t, with all graphs on one coordinate system.
Exercise 2.
Repeat Exercise 1. For the damped case, with γ = .1 and 0 ≤ t ≤20 the same initial conditions.
Exercise 3.
By trial and error, find the initial condition of the form (x(0), y(0) = (0, y_{o}) so that x(t) → π as t → ∞
Plot your attempts in the x,y plane as in Exercise 2a., and turn in a plot using your best guess.
Note: When you are finished, you should have your plots on exactly seven different coordinate systems: three each for Exercises 1 and 2, and one for Exercise 3.