Consider the initial-value problem d 2 u dt 2 sinu 0, u(0) p 12, u9(0) 1 3 for the

Chapter 3, Problem 24

(choose chapter or problem)

Consider the initial-value problem

\(\frac{d^{2} \theta}{d t^{2}}+\sin \theta=0\), \(\theta(0)=\frac{\pi}{12}\), \(\theta^{\prime}(0)=-\frac{1}{3}\)

for the nonlinear pendulum. Since we cannot solve the differential equation, we can find no explicit solution of this problem. But suppose we wish to determine the first time \(t_{1}>0\) for which the pendulum in Figure 3.11.3, starting from its initial position to the right, reaches the position OP-that is, find the first positive root of \(\theta(t)=0\). In this problem and the next we examine several ways to proceed.

(a) Approximate \(t_{1}\) by solving the linear problem

\(\frac{d^{2} \theta}{d t^{2}}+\theta=0\), \(\theta(0)=\frac{\pi}{12}\), \(\theta^{\prime}(0)=-\frac{1}{3}\)

(b) Use the method illustrated in Example 3 of Section 3.7 to find the first four nonzero terms of a Taylor series solution \(\theta(t)\) centered at 0 for the nonlinear initial-value problem. Give the exact values of all coefficients.

(c) Use the first two terms of the Taylor series in part (b) to approximate \(t_{1}\).

(d) Use the first three terms of the Taylor series in part (b) to approximate \(t_{1}\).

(e) Use a root-finding application of a CAS (or a graphing calculator) and the first four terms of the Taylor series in part (b) to approximate \(t_{1}\).

(f) In this part of the problem you are led through the commands in Mathematica that enable you to approximate  the root \(t_{1}\). The procedure is easily modified so that any root of \(\theta(t)=0\) can be approximated. (If you do not have Mathematica, adapt the given procedure by finding the corresponding syntax for the CAS you have on hand.) Reproduce and then, in turn, execute each line in the given sequence of commands.

(g) Appropriately modify the syntax in part (f) and find the next two positive roots of \(\theta(t)=0\).