Suppose the circuit described by the equation below

$$\frac{dI}{dt}+\frac{R}{L}I= \frac{1}{L}V(t)$$

is driven by a sinusoidal voltage source $V(t)=V \sin \omega t$ (where $V$ and $\omega$ are constant).

(a) Show that

$$I(t)=\frac{V}{R^2+L^2 \omega^2}(R \sin \omega t-L \omega \cos \omega t)+C e^{-(R / L) t}$$

(b) Let $Z=\sqrt{R^2+L^2 \omega^2}$. Choose $\theta$ so that $Z \cos \theta=R$ and $Z \sin \theta=$ $L \omega$. Use the addition formula for the sine function to show that

$$I(t)=\frac{V}{Z} \sin (\omega t-\theta)+C e^{-(R / L) t}$$

This shows that the current in the circuit varies sinusoidally apart from a DC term (called the transient current in electronics) that decreases exponentially.

A rabbit population on an island increases exponentially with growth rate

$$k = 0.12 \text { months } ^ { - 1 }.$$

When the population reaches 300 rabbits (say, at time t = 0), wolves begin eating the rabbits at a rate of r rabbits per month. (a) Find a differential equation satisfied by the rabbit population P(t). (b) How large can r be without the rabbit population becoming extinct?

A tank has the shape of the parabola $y=x^2$, revolved around the y-axis. Water leaks from a hole of area B=0.0005 $m^2$ at the bottom of the tank. Let y(t) be the water level at time t. How long does it take for the tank to empty if it is initially filled to height $y_0$=1 m ?

In Exercise, use the differential equation for a leaking container, Eq. $\frac{d y}{d t}=-\frac{B \sqrt{2 g y}}{A(y)}$

The tank in Figure is a cylinder of radius 4 m and height 15 m. Assume that the tank is half-filled with water and that water leaks through a hole in the bottom of area $B=0.001 \mathrm{~m}^2$. Determine the water level y(t) and the time $t_e$ when the tank is empty.

In this exercise, let $\alpha(x)$ be an integrating factor for $y^{\prime}+P(x) y=Q(x)$. The differential equation $y^{\prime}+P(x) y=0$ is called the associated homogeneous equation.

(a) Show that $y=1 / \alpha(x)$ is a solution of the associated homogeneous equation.

(b) Show that if $y=f(x)$ is a particular solution of $y^{\prime}+P(x) y=Q(x)$, then $f(x)+C / \alpha(x)$ is also a solution for any constant $C$.

A lake has a carrying capacity of 1000 fish. Assume that the fish population grows logistically with growth constant

$$k = 0.2 \mathrm { day } ^ { - 1 }.$$

How many days will it take for the population to reach 900 fish if the initial population is 20 fish?

In Exercise, use the differential equation for a leaking container, Eq. $\frac{d y}{d t}=-\frac{B \sqrt{2 g y}}{A(y)}$

At t=0, a conical tank of height 300 cm and top radius 100 cm [Figure] is filled with water. Water leaks through a hole in the bottom of area $B=3 \mathrm{~cm}^2$. Let y(t) be the water level at time t.

(a) Show that the tank's cross-sectional area at height y is $A(y)=\frac{\pi}{9} y^2$.

(b) Solve the differential equation satisfied by y(t)

(c) How long does it take for the tank to empty?

Assume that the outside temperature varies as

T(t)=15+5 sin ($\pi$ t / 12)

where $t=0$ is 12 noon. A house is heated to $25^{\circ}$C at t=0 and after that, its temperature y(t) varies according to Newton's Law of Cooling :

dy/dt=-0.1(y(t)-T(t))

Use Earlier Exercise to solve for y(t).

Suppose that y' = ky(1 - y/8) has a solution satisfying y(0) = 12 and y(10) = 24. Find k.

In Exercise, use the differential equation for a leaking container, Eq. $\frac{d y}{d t}=-\frac{B \sqrt{2 g y}}{A(y)}$

Water leaks through a hole of area $B=0.002 \mathrm{~m}^2$ at the bottom of a cylindrical tank that is filled with water and has height 3 m and a base of area $10 \mathrm{~m}^2$. How long does it take (a) for half of the water to leak out and (b) for the tank to empty ?

In this exercise, let a, b, r be constants. Show that

$$y=C e^{-k t}+a+b k\left(\frac{k \sin r t-r \cos r t}{k^2+r^2}\right)$$

is a general solution of

$$\frac{d y}{d t}=-k(y-a-b \sin r t)$$

In this exercise, let y(t) be the solution of dy/dt=0.3y(2-y) with y(0)=1. Determine $\lim \limits_{t \to \infty}$ y(t) without solving for y explicitly.

Continuing with the previous exercise, let tank 2 be another tank filled with V$_2$ gallons of water. Assume that the dye solution from tank 1 empties into tank 2 as in the previous figure, mixes instantaneously, and leaves tank 2 at the same rate R. Let c$_2$(t) be the dye concentration in tank 2 at time t.

(a) Explain why c$_2$ satisfies the differential equation

$$\frac{d c_2}{d t}=\frac{R}{V_2}\left(c_1-c_2\right)$$

(b) Use the solution to the previous exercise to solve for $c_2(t)$ if $V_1=300$, $V_2=200, R=50$, and $c_0=10$.

(c) Find the maximum concentration in tank 2.

(d) Plot the solution.

In the earlier example, we calculated the thickness of a glacier assuming the friction at the base of the glacier is constant. In this exercise, we consider cases where the friction varies along the length of the glacier.

(a) Solve the glacier thickness differential equation, and use the equation

$$T \frac{d T}{d x}=\frac{\tau}{\rho g}$$

for T(x) with $\tau(x)=0.3 x(1000-x)$ N/m$^2$ and initial condition T(0)=0.

(b) Sketch the graph of T for $0 \leq x \leq 1000$ m.