The given differential equation has a fundamental set of solutions whose Wronskian $W(t)$ is such that $W(0)=1$. What is $W(4)$ ?

$$y^{(4)}-y^{\prime \prime}+y=0$$

The function $y(t)$ is a solution of the initial value problem $y^{\prime \prime}+a y^{\prime}+b y=0, y\left(t_0\right)=y_0$, $y^{\prime}\left(t_0\right)=y_0^{\prime}$, where the point $t_0$ is specified. Determine the constants $a, b, y_0$, and $y_0^{\prime}$.

$$y(t)=2 \sin 2 t+\cos 2 t, \quad t_0=\pi / 4$$

In each exercise, (a) Find the general solution of the differential equation. (b) If initial conditions are specified, solve the initial value problem.

$$y^{(4)}-y^{\prime \prime \prime}+y^{\prime}-y=0$$

In these exercises, the $t$-interval of interest is $-\infty<t<\infty$ unless indicated otherwise. (a) Verify that the given functions are solutions of the differential equation. (b) Calculate the Wronskian. Do the two functions form a fundamental set of solutions? (c) If the two functions form a fundamental set, determine the unique solution of the initial value problem.

$$4 y^{\prime \prime}+4 y^{\prime}+y=0 ; \quad y_1(t)=e^{-t / 2}, \quad y_2(t)=t e^{-t / 2} ; \quad y(1)=1, \quad y^{\prime}(1)=0$$

For the given differential equation, (a) Determine the complementary solution, $y_C(t)=c_1 y_1(t)+c_2 y_2(t)$. (b) Use the method of variation of parameters to construct a particular solution. Then form the general solution.

$y^{\prime \prime}-\left[2+(2 / t)\left|y^{\prime}+\right| 1+(2 / t)\right] y=e^t, \quad 0<t<\infty$. [The function $y_1(t)=e^t$ is a solution of the homogeneous equation.]

In this problem, we explore computationally the question posed in Exercise 12. Consider the initial value problem

$$y^{\prime \prime}+\gamma y^{\prime}+y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 .$$

where, for simplicity, we have given the mass, spring constant, and initial displacement all a numerical value of unity. (a) Determine $\gamma_{\text {crit }}$ the damping constant value that makes the given spring-massdashpot system critically damped. (b) Use computational software to plot the solution of the initial value problem for $\gamma=\gamma_{\text {crit }}, 2 \gamma_{\text {crit, }}$ and $20 \gamma_{\text {crit }}$ over a common time interval sufficiently large to display the main features of each solution. What trend do you observe in the behavior of the solutions as $\gamma$ increases? Is it consistent with the conclusions reached in Exercise 12?

The graph of a solution $y(t)$ of the differential equation $4y^{\prime\prime} + 4y^\prime + y = 0$ passes through the points $(1, e^{1/2)}$ and $(2, 0)$. Determine $y(0)$ and $y^\prime(0)$.

In each exercise, determine the general solution. If initial conditions are given, solve the initial value problem.

$$y^{\prime \prime}+2 y^{\prime}+y=8, \quad y(0)=10, \quad y^{\prime}(0)=1$$

Since $\sin (\omega t+2 \pi)=\sin \omega t$ and $\cos (\omega t+2 \pi)=\cos \omega t$, the amount of time $T$ it takes a bobbing object to go through one cycle of its motion is determined by the relation $\omega T=2 \pi$, or $T=2 \pi / \omega$. This time $T$ is called the period of the motion (see Section 3.6). As the period decreases, the bobbing motion of the floating object becomes more rapid.

(a) Two identically shaped cylindrical drums, made of different material, are floating at rest as shown in part (a) of the figure. (b) Two cylindrical drums, made of identical material, are floating at rest as shown in part (b). For each case, when the drums are put into motion, is it possible to identify the drum that will bob up and down more rapidly? Explain.

The function $y(t)$ is a solution of the initial value problem $y^{\prime \prime}+a y^{\prime}+b y=0, y\left(t_0\right)=y_0$, $y^{\prime}\left(t_0\right)=y_0^{\prime}$, where the point $t_0$ is specified. Determine the constants $a, b, y_0$, and $y_0^{\prime}$.

$$y(t)=\sin t-\sqrt{2} \cos t, \quad t_0=\pi / 4$$

In each exercise, (a) Find the general solution of the differential equation. (b) If initial conditions are specified, solve the initial value problem.

$$y^{(6)}-y=0$$

For the given differential equation, (a) Determine the complementary solution, $y_C(t)=c_1 y_1(t)+c_2 y_2(t)$. (b) Use the method of variation of parameters to construct a particular solution. Then form the general solution.

$(t-1)^2 y^{\prime \prime}-4(t-1) y^{\prime}+6 y=t, \quad 1<t<\infty$. [The function $y_1(t)=(t-1)^2$ is a solution of the homogeneous equation.]

For the given differential equation, (a) Determine the complementary solution. (b) Use the method of undetermined coefficients to find a particular solution. (c) Form the general solution.

$$9 y^{\prime \prime}-6 y^{\prime}+y=9 t e^{t / 3}$$

The Great Zacchini, a daredevil extraordinaire, is a circus performer whose act consists of being "shot from a cannon" to a safety net some distance away. The "cannon" is a frictionless tube containing a large spring, as shown in the figure. The spring constant is $k=150 \mathrm{lb} / \mathrm{ft}$, and the spring is pre-compressed $10 \mathrm{ft}$ prior to launching the acrobat. Assume that the spring obeys Hooke's law and that Zacchini weighs $150 \mathrm{lb}$. Neglect the weight of the spring. (a) Let $x(t)$ represent spring displacement along the tube axis, measured positive in the upward direction. Show that Newton's second law of motion leads to the differential equation $m x^{\prime \prime}=-k x-m g \cos (\pi / 4), x<0$, where $m$ is the mass of the daredevil. Specify appropriate initial conditions. (b) With what speed does he emerge from the tube when the spring is released? (c) If the safety net is to be placed at the same height as the mouth of the "cannon," how far downrange from the cannon's mouth should the center of the net be placed?