Chapter 5.3, Problem 20E

In Problems 14-24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge-Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)

Appendix G describes various websites and commercial software that sketch direction fields and automate most of the differential equation algorithms discussed in this book.

In Project C of Chapter 4, it was shown that the simple pendulum equation

${{\theta }^{\prime }}^{\prime }\left(t\right)+\sin \theta \left(t\right)=0$

has periodic solutions when the initial displacement and velocity are small. Show that the period of the solution may depend on the initial conditions by using the vectorized Runge-Kutta algorithm with $h=0.02$ to approximate the solutions to the simple pendulum problem on $\left[0,4\right]$ for the initial conditions:

a. $\theta \left(0\right)=0.1$, ${\theta }^{\prime }\left(0\right)=0$.

b. $\theta \left(0\right)=0.5$, ${\theta }^{\prime }\left(0\right)=0$.

c. $\theta \left(0\right)=1.0$, ${\theta }^{\prime }\left(0\right)=0$.

[Hint: Approximate the length of time it takes to reach $-\theta \left(0\right)$.]