Consider the initial value problemu + u + u = 0, u(0) = 2, u(0) = 0.We wish to explore

Chapter 3, Problem 25

(choose chapter or problem)

Consider the initial value problemu + u + u = 0, u(0) = 2, u(0) = 0.We wish to explore how long a time interval is required for the solution to become negligibleand how this interval depends on the damping coefficient . To be more precise,let us seek the time such that |u(t)| < 0.01 for all t > . Note that critical damping forthis problem occurs for = 2.(a) Let = 0.25 and determine , or at least estimate it fairly accurately from a plot ofthe solution.(b) Repeat part (a) for several other values of in the interval 0 << 1.5. Note that steadily decreases as increases for in this range.(c) Create a graph of versus by plotting the pairs of values found in parts (a) and (b).Is the graph a smooth curve?(d) Repeat part (b) for values of between 1.5 and 2. Show that continues to decreaseuntil reaches a certain critical value 0, after which increases. Find 0 and thecorresponding minimum value of to two decimal places.(e) Another way to proceed is to write the solution of the initial value problem inthe form (26). Neglect the cosine factor and consider only the exponential factor and theamplitude R. Then find an expression for as a function of . Compare the approximateresults obtained in this way with the values determined in parts (a), (b), and (d).