Chapter 8.2, Problem 21P

Consider the initial value problem

$\begin{array}{cc}y\text{'}=\cos 5\pi t,& y(0)=1\end{array}$.

a) Determine the solution $y=\varphi (t)$ and draw a graph of $y=\varphi (t)$ for $0\le t\le 1$.

b) Determine approximate values of $\varphi (t)$ at $t=0.2,0.4,$ and $0.6$ using the Euler method with $h=0.2$. Draw a broken-linegraph for the approximate solution and compare it with the graph of the exact solution.

c) Repeat the computation of part (b) for $0\le t\le 0.4$, but take $h=0.1$.

d) Show by computing the local truncation error that neither of these step sizes is sufficiently small. Determine avalue of $h$ to ensure that the local truncation error is less than $0.05$ throughout the interval $0\le t\le 1$. That such a smallvalue of $h$ is required results from the fact that max $\left|\varphi \text{'}\text{'}(t)\right|$ islarge.