Consider the initial value problem y? = cos 5?t, y(0) =
Question and Solution
Consider the initial value problem y? = cos 5?t, y(0) = 1. (a) Determine the solution y = ?(t) and draw a graph of y = ?
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Consider the initial value problem y? = cos 5?t, y(0) = 1. (a) Determine the solution y = ?(t) and draw a graph of y = ?(t) for 0 ? t ? 1. (b) Determine approximate values of ?(t) at t = 0.2, 0.4, and 0.6 using the Euler method with h = 0.2. Draw a broken-line graph for the approximate solution and compare it with the graph of the exact solution. (c) Repeat the computation of part (b) for 0 ? t ? 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 a value of h to ensure that the local truncation error is less than 0.05 throughout the interval 0 ? t ? 1. That such a small value of h is required results from the fact that max |???(t)| is large.
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