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

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

In Figure $11.3$, the height, $y$, of the hanging cable above the horizontal line satisfies

$$\frac{d^{2}y}{dx^{2}}=k\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}.$$

(a) Show that $y=\frac{e^{x}+e^{-x}}{2}$ satisfies this differential equation if $k=1.$

Look at the pendulum motion in the previous problem. What effect does it have on $x$ as a function of time if:
$l$ is increased?

Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 1 million barrels of oil in the well; six years later 500,000 barrels remain. At what rate was the amount of oil in the well decreasing when there were 600,000 barrels remaining?

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.

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?

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 $1950$ .)

Find the value(s) of $\omega$ for which $y=\cos\omega t$ satisfies

$$\frac{d^{2}y}{dt^{2}} + 9y=0.$$

Look at the pendulum motion in the previous problem. What effect does it have on $x$ as a function of time if:
$x_0$ is increased?

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

$\frac{d y}{d x}=\frac{5 y}{x}, \quad y=3$ where $x=1$

Analyze the phase plane of the differential equations for $x, y \geq 0.$ Show the nullclines and equilibrium points, and sketch the direction of the trajectories in each region.

$$\begin{align*} &\frac{dx}{dt}= x(1-y-\frac{x}{3})\\ &\frac{dy}{dt}= y(1-\frac{y}{2}-x) \end{align*}$$

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.)
Using Lanchester's square law derived in th previous problem, find the equation of the trajectory describing the battle.

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. Let y=f(t) be the fraction of the original material remembered t weeks after the course has ended. Set up a differential equation for y. Your equation will contain two constants; the constant a is less than y for all t.

Determine which of the following differential equations are separable. Do not solve the equations.

(l) $y^{\prime}=(x+y) /(x+2 y)$