Exercise 1. 

Graph several solutions to dx/dt = y, dy/dt = -k² 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.