Chapter 4.4, Problem 25P

Consider the initial value problem $\begin{array}{ccc}y\text{'}\text{'}+\gamma y\text{'}+y=0,& y\left(0\right)=2,& y\text{'}\left(0\right)=0\end{array}$. We wish to explore how long a time interval is required for the solution to become “negligible” and how this interval depends on the damping coefficient $\gamma$. To be more precise, let us seek the time $\tau$. Note that critical damping for this problem occurs for $\gamma =2$.

Let $\gamma =0.25$ and determine $\tau$, or at least estimate it fairly accurately from a plot of the solution.

Repeat part (a) for several other values of $\gamma$ in the interval $0<\gamma <1.5$. Note that $\tau$ steadily decreases as $\gamma$ increases for $\gamma$ in this range.

Create a graph of $\tau$ versus $\gamma$ by plotting the pairs of values found in parts (a) and (b). Is the graph a smooth curve?

Repeat part (b) for values of $\gamma$ between $1.5$ and $2$. Show that $\tau$ continues to decrease until $\gamma$ reaches a certain critical value ${\gamma }_0$, after which $\tau$ increases. Find ${\gamma }_0$ and the corresponding minimum value of $\tau$ to two decimal places.

Another way to proceed is to write the solution of the initial value problem in the form $\left(23\right)$. Neglect the cosine factor and consider only the exponential factor and the amplitude $R$. Then find an expression for $\tau$ as a function of $\gamma$. Compare the approximate results obtained in this way with the values determined in parts (a), (b), and (d).