Numerically solve the differential equation y'' + sin(y) = 0 using initial conditions y(0)= 0, y′(0) = 1. Solve equation y'' + y = 0 with the same initial conditions. Plot on the same graph the solutions to both the nonlinear equation (first) and the linear equation (second) on the interval from t = 0 to t = 40, and compare the two. Be clear about which curve is the nonlinear solution and which is the linear solution.
Compare the linear and nonlinear solutions for each of the following values of the initial velocity v: 1, 1.99, 2, 2.01. For the (numerical) nonlinear solution, interpret what the graph indicates the pendulum is doing physically. What do you think the exact solution does in each case?