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Problem 21

Fluid Ejection. In the design of a sewage treatme…

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Problem 20

In Project C of Chapter 4, it was shown that the simple pendulum equation
$$\theta^{\prime \prime}(t)+\sin \theta(t)=0$$
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 [0, 4] for the initial conditions:
$$\begin{array}{l}{\text { (a) } \theta(0)=0.1, \quad \theta^{\prime}(0)=0} \\ {\text { (b) } \theta(0)=0.5, \quad \theta^{\prime}(0)=0} \\ {\text { (c) } \theta(0)=1.0,} \\ {[\text { Hint: Approximate the length of time it takes to reach }} \\ {-\theta(0) .}\end{array}$$

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