Chapter 3: Problem 10
The Wronskian of two functions is \(W(t)=t^{2}-4 .\) Are the functions linearly independent or linearly dependent? Why?
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Find the solution of the initial value problem
$$
u^{\prime \prime}+u=F(t), \quad u(0)=0, \quad u^{\prime}(0)=0
$$
where
$$
F(t)=\left\\{\begin{array}{ll}{F_{0}(2 \pi-t),} & {0 \leq t \leq \pi} \\ {-0}
& {(2 \pi-t),} & {\pi
Consider the initial value problem $$ u^{\prime \prime}+\gamma u^{\prime}+u=0, \quad u(0)=2, \quad u^{\prime}(0)=0 $$ 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 \(\tau\) such that \(|u(t)|<0.01\) for all \(t>\tau .\) Note that critical damping for this problem occurs for \(\gamma=2\) (a) Let \(\gamma=0.25\) and determine \(\tau,\) or at least estimate it fairly accurately from a plot of the solution. (b) 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. (c) Obtain a graph of \(\tau\) versus \(\gamma\) 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 \(\gamma\) 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. (e) Another way to proceed is to write the solution of the initial value problem in the form (26). 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).
In each of Problems 13 through 18 find the solution of the given initial value problem. $$ y^{\prime \prime}+y^{\prime}-2 y=2 t, \quad y(0)=0, \quad y^{\prime}(0)=1 $$
In each of Problems 1 through 12 find the general solution of the given differential equation. $$ y^{\prime \prime}-2 y^{\prime}-3 y=3 e^{2 x} $$
In many physical problems the nonhomogencous term may be specified by different formulas in different time periods. As an example, determine the solution \(y=\phi(t)\) of $$ y^{\prime \prime}+y=\left\\{\begin{array}{ll}{t,} & {0 \leq t \leq \pi} \\\ {\pi e^{x-t},} & {t>\pi}\end{array}\right. $$ $$ \begin{array}{l}{\text { satisfying the initial conditions } y(0)=0 \text { and } y^{\prime}(0)=1 . \text { Assume that } y \text { and } y^{\prime} \text { are also }} \\ {\text { continuous at } t=\pi \text { . Plot the nonhomogencous term and the solution as functions of time. }} \\ {\text { Hint: First solve the initial value problem for } t \leq \pi \text { ; then solve for } t>\pi \text { , determining the }} \\ {\text { constants in the latter solution from the continuity conditions at } t=\pi \text { . }}\end{array} $$
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