Assume that you fish for 3 years, then fishing is banned for the next 3 years. Thereafter you start again. And so on. This is called intermittent harvesting. Describe qualitatively how the population will develop if intermitting is continued periodically. Find and graph the solution for the first 9 years, assuming that A=B=1, H=0.2, and y(0)=2.

Lake Erie has a water volume of about 450 km³ and a flow rate (in and out) of about 175 km² per year. If at some instant the lake has pollution concentration p=0.04%, how long, approximately, will it take to decrease it to p/2, assuming that the inflow is much cleaner, say, it has pollution concentration p/4, and the mixture is uniform (an assumption that is only imperfectly true)? First guess.

Heating and cooling of a building can be modeled by the ODE T'=k1(T-Ta)+k(T-Tω)+P, where T=T(t) is the temperature in the building at time t, Ta the outside temperature, Tω the temperature wanted in the building, and P the rate of increase of T due to machines and people in the building, and k1 and k2 are (negative) constants. Solve this ODE, assuming P=const, Tω=const, and Ta varying sinusoidal over 24 hours, say, Ta=A-C cos(2π/24)t. Discuss the effect of each term of the equation on the solution.

If a wet sheet in a dryer loses its moisture at a rate proportional to its moisture content, and if it loses half of its moisture during the first 10 min of drying, when will it be practically dry, say, when will it have lost 99% of its moisture? First guess, then calculate.

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y'+2.5y=1.6x

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y'-0.4y=29 sin x

Solution curves of $y'=g(y/x)$. Show that any (nonvertical) straight line through the origin of the $xy$-plane intersects all these curves of a given ODE at the same angle.

If the isotherms in a region are $x^2-y^2=c$ what are the curves of heat flow (assuming orthogonality)?

In a room containing 20,000 ft³ of air, 600 ft³ of fresh air flows in per minute, and the mixture (made practically uniform by circulating fans) is exhausted at a rate of 600 cubic feet per minute (cfm). What is the amount of fresh air y(t) at any time if y(0)=0? Afther what time will 90% of the air be fresh?

Suppose that the tank is hemispherical, of radius $R$, initially full of water, and has an outlet of $5 \mathrm{~cm}^2$ crosssectional area at the bottom. (Make a sketch.) Set up the model for outflow. Indicate what you can use (so that it can become part of the general method independent of the shape of the tank). Find the time $t$ to empty the tank (a) for any $R$, (b) for $R=1 \mathrm{~m}$. Plot $t$ as function of $R$. Find the time when $h=R / 2$ (a) for any $R$, (b) for $R=1 \mathrm{~m}$.

Two forces act on a parachutist, the attraction by the earth mg (m=mass of person plus equipment, g=9,8 m/sec² the acceleration of gravity) and the air resistance, assumed to be proportional to the square of the velocity v(t). Using Newton’s second law of motion (mass X acceleration = resultant of the forces), set up a model (an ODE for v(t)). Graph a direction field (choosing mand the constant of proportionality equal to 1). Assume that the parachute opens when v=10 m/sec. Graph the corresponding solution in the field. What is the limiting velocity? Would the parachute still be sufficient if the air resistance were only proportional to v(t)?

Suppose that the population $y(t)$ of a certain kind of fish is given by the logistic equation , and fish are caught at a rate $H y$ proportional to $y$. Solve this so-called Schaefer model. Find the equilibrium solutions $y_1$ and $y_2(>0)$ when $H<A$. The expression $Y=H y_2$ is called the equilibrium harvest or sustainable yield corresponding to $H$. Why? Find and graph the solution satisfying $y(0)=2$ when (for simplicity) $A=B=1$ and $H=0.2$. What is the limit? What does it mean? What if there were no fishing?

Jack, arrested when leaving a bar, claims that he has been inside for at least half an hour (which would provide him with an alibi). The police check the water temperature of his car (parked near the entrance of the bar) at the instant of arrest and again 30 min later, obtaining the values 190°F and 110°F, respectively. Do these results give Jack an alibi? (Solve by inspection.)

An amplifier with an input resistance of $100 \mathrm{k} \Omega$ and an output resistance of $1 \mathrm{k} \Omega$ is to be capacitor-coupled to a $10-\mathrm{k} \Omega$ source and a $1-\mathrm{k} \Omega$ load. Available capacitors have values only of the form $1 \times 10^{-n} \mathrm{F}.$ What are the values of the smallest capacitors needed to ensure that the corner frequency associated with each is less than 100 Hz? What actual corner frequencies result? For the situation in which the basic amplifier has an open-circuit voltage gain $\left(A_{v o}\right)$ of 100 V/V, find an expression for $T(s)=V_{o}(s) / V_{s}(s).$