Draw the error bounds shown in Figure. See Exercise for assistance.
Let’s plot the error bounds shown in Figure 1. First, solve x' = (x – 1) cos t, x(0) = 0, and plot the solution over the interval [–4, 4). Next, as we saw in Example 8.1, if y(t) is a second solution with |x(0) – y(0)| ≤ 0.1, then the inequality (8.3) becomes |x(t) – y(t)| ≤ 0.1e|t|. Solve this inequality for x(t), placing your final answer in the form eL(t) ≤ x(t) ≤ eH(t) Then add the graphs of eL(t) and eH(t) to your plot. How can you use Theorem to show that no solution starting with initial condition |x(0) – y(0)| ≤ 0.1 has any chance of rising as far as indicated by eH (t)?
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