The population of a species of elk on Reading Island in Canada has been monitored for some years. When the population was $6(00$, the relative birth rate was $35 \%$ and the relative death rate was $15 \%$. As the population grew to $800 .$ the corresponding figures were $30 \%$ and $20 \%$. The island is isolated so there is no hunting or migration.
Find the equilibrium size of the population. Today there are $900$ elk on Reading Island. How do you expect the population to change in the future?

A detective finds a murder victim at 9 am. The temperature of the body is measured at $90.3^{\circ} \mathrm{F}$. One hour later, the temperature of the body is $89.0^{\circ} \mathrm{F}$. The temperature of the room has been maintained at ta constant $68^{\circ} \mathrm{F}$. Assuming the temperature, T, of the body obeys Newton's Law of Cooling, write a differential equation for T.

An item is initially sold at a price of $\$ p$ per unit. Over time, market forces push the price toward the equilibrium price, $\$ p^*$, at which supply balances demand. The Evans Price Adjustment model says that the rate of change in the market price, $\$ p$, is proportional to the difference between the market price and the equilibrium price.

(a) Write a differential equation for $p$ as a function of $t$.

The population of a species of elk on Reading Island in Canada has been monitored for some years. When the population was $6(00$, the relative birth rate was $35 \%$ and the relative death rate was $15 \%$. As the population grew to $800 .$ the corresponding figures were $30 \%$ and $20 \%$. The island is isolated so there is no hunting or migration.
Write a differential equation to model the population as a function of time. Assume that relative growth rate is a linear function of population.

Expand $A \sin (\omega t+\phi)$ using the trigonometric identity $\sin (x+y)=\sin x \cos y+\cos x \sin y$.

Solve the boundary value problem. \

$$p^{\prime \prime}+4 p^{\prime}+5 p=0 . \quad p(0)=1, \quad p(\pi / 2)=5$$

Suppose that $y=f(x)$ is a solution of the differential equation $d y / d x=g(x)$. Are the statements in the following Problems true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.
If $g(x)$ is even, then $f(x)$ is odd.

For the differential equation, find a solution which passes through the given point.

$$e^{-\cos \theta} \frac{d z}{d \theta}=\sqrt{1-z^2} \sin \theta, \quad z(0)=\frac{1}{2}$$

Use separation of variables to find the solutions to the differential equations subject to the given initial conditions.

$\frac{d z}{d t}=z+z t^2, \quad z=5$ when $t=0$

As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, a. Describe the practical meaning (in terms of the amount remembered) of the constants in the solution y=f(t).

The total number of people infected with a virus often grows like a logistic curve. Suppose that time, $t$, is in weeks and that $10$ people originally have the virus. In the early stages, the number of people infected is increasing exponentially with $k=1.78$. In the long run, $5000$ people are infected.

(c ) Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point?

The amount of land in use for growing crops increases as the world's population increases. Suppose $A(t)$ represents the total number of hectares of land in use in year t. (A hectare is about $2 \dfrac{1}{2}$ acres.)
In $1950$ about $1 \cdot 10^9$ hectares of land were in use; in $1980$ the figure was $2 \cdot 10^{\circ}$. If the total amount of land available for growing crops is thought to be $3.2 \cdot 10^{17}$ hectares, when does this model predict it is exhausted? (Let $t=0$ in $1950$ .)

Consider a conflict between two armies of x and y soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and $t$ represents time since the start of the battle, then $x$ and $y$ obey the differential equations

$$\begin{aligned} & \frac{d x}{d t}=-a y \\ & \frac{d y}{d t}=-b x \quad a, b>0 . \end{aligned}$$

Consider the battle of Iwo Jima, described in Problem 21. Take $a=0.05, b=0.01$ and assume the initial strength of the US troops to be 54,000 and that of the Japanese troops to be 21,500. (Again, ignore reinforcements.)
Assuming that the Japanese fought without surrendering until they had all been killed, as was the case, how many US troops does this model predict would be left when the battle ended?

As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, a.

Solve the differential equation.