Chapter 10

A quantity \(W\) satisfies the differential equation $$ \frac{d W}{d t}=5 W-20 $$ (a) Is \(W\) increasing or decreasing at \(W=10 ? W=2\) ? (b) For what values of \(W\) is the rate of change of \(W\) equal to zero?

A patient is given the drug theophylline intravenously at a rate of \(43.2 \mathrm{mg} /\) hour to relieve acute asthma. The rate at which the drug leaves the patient's body is proportional to the quantity there, with proportionality constant \(0.082\) if time, \(t\), is in hours. The patient's body contains none of the drug initially. (a) Describe in words how you expect the quantity of theophylline in the patient to vary with time. (b) Write a differential equation satisfied by the quantity of theophylline in the body, \(Q(t)\). (c) Solve the differential equation and graph the solution. What happens to the quantity in the long run?

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. (a) At what rate was the amount of oil in the well decreasing when there were 600,000 barrels remaining? (b) When will there be 50,000 barrels remaining?

One theory on the speed an employee learns a new task claims that the more the employee already knows, the more slowly he or she learns. Suppose that the rate at which a person learns is equal to the percentage of the task not yet learned. If \(y\) is the percentage learned by time \(t\), the percentage not yet learned by that time is \(100-y\), so we can model this situation with the differential equation $$ \frac{d y}{d t}=100-y $$ (a) Find the general solution to this differential equation. (b) Sketch several solutions. (c) Find the particular solution if the employee starts learning at time \(t=0\) (so \(y=0\) when \(t=0\) ).

In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For example, this is the case as \(\delta\) glucono-lactone changes into gluconic acid. (a) Write a differential equation satisfied by \(y\), the quantity of \(\delta\) -glucono-lactone present at time \(t\). (b) If 100 grams of \(\delta\) -glucono-lactone is reduced to \(54.9\) grams in one hour, how many grams will remain after 10 hours?

An early model of the growth of the Wikipedia assumed that every day a constant number, \(B\), of articles are added by dedicated wikipedians and that other articles are created by the general public at a rate proportional to the number of articles already there. Express this model as a differential equation for \(N(t)\), the total number of Wikipedia articles \(t\) days after January 1, 2001 .

Let \(w\) be the number of worms (in millions) and \(r\) the number of robins (in thousands) living on an island. Suppose \(w\) and \(r\) satisfy the following differential equations, which correspond to the slope field in Figure \(10.42\). $$ \frac{d w}{d t}=w-w r, \quad \frac{d r}{d t}=-r+w r $$ Assume that \(w=3\) and \(r=1\) when \(t=0\). Do the numbers of robins and worms increase or decrease initially? What happens in the long run?

A chain smoker smokes five cigarettes every hour. From cach cigarette, \(0.4 \mathrm{mg}\) of nicotine is absorbed into the person's bloodstream. Nicotine leaves the body at a rate proportional to the amount present, with constant of proportionality \(-0.346\) if \(t\) is in hours. (a) Write a differential equation for the level of nicotine in the body, \(N\), in \(\mathrm{mg}\), as a function of time, \(t\), in hours. (b) Solve the differential equation from part (a). Initially there is no nicotine in the blood. (c) The person wakes up at 7 am and begins smoking. How much nicotine is in the blood when the person goes to sleep at \(11 \mathrm{pm}(16\) hours later \() ?\)

Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body. The half-life of hydrocodone bitartrate in the body is \(3.8\) hours and the dose is \(10 \mathrm{mg}\). (a) Write a differential equation for the quantity, \(Q\), of hydrocodone bitartrate in the body at time \(t\), in hours since the drug was fully absorbed. (b) Solve the differential equation given in part (a). (c) Use the half-life to find the constant of proportionality, \(k\). (d) How much of the \(10-\mathrm{mg}\) dose is still in the body after 12 hours?

A country's infrastructure is its transportation and communication systems, power plants, and other public institutions. The Solow model asserts that the value of national infrastructure \(K\) increases due to investment and decreases due to capital depreciation. The rate of increase due to investment is proportional to national income, \(Y\). The rate of decrease due to depreciation is proportional to the value of existing infrastructure. Write a differential equation for \(K\).

Let \(w\) be the number of worms (in millions) and \(r\) the number of robins (in thousands) living on an island. Suppose \(w\) and \(r\) satisfy the following differential equations, which correspond to the slope field in Figure \(10.42\). $$ \frac{d w}{d t}=w-w r, \quad \frac{d r}{d t}=-r+w r $$ At \(t=0\) there are \(2.2\) million worms and 1 thousand robins. (a) Use the differential equations to calculate the derivatives \(d w / d t\) and \(d r / d t\) at \(t=0\). (b) Use the initial values and your answer to part (a) to estimate the number of robins and worms at \(t=0.1\). (c) Using the method of part (a) and (b), estimate the number of robins and worms at \(t=0.2\) and \(0.3\).

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, \(a\). (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 \(t\). (b) Solve the differential equation. (c) Describe the practical meaning (in terms of the amount remembered) of the constants in the solution \(y=f(t) .\)

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 \(2 \frac{1}{2}\) acres.) (a) Explain why it is plausible that \(A(t)\) satisfies the equation \(A^{\prime}(t)=k A(t) .\) What assumptions are you making about the world's population and its relation to the amount of land used? (b) In 1950 about \(1 \cdot 10^{9}\) hectares of land were in use; in 1980 the figure was \(2 \cdot 10^{9} .\) If the total amount of land available for growing crops is thought to be \(3.2 \cdot 10^{9}\) hectares, when does this model predict it is exhausted? (Let \(t=0\) in \(1950 .\) )

If the initial population of fish is 70 million, use the differential equation \(d P / d t=0.2 P-10\) to estimate the fish population after \(1,2,3\) years.

(a) What are the equilibrium solutions for the differential equation $$ \frac{d y}{d t}=0.2(y-3)(y+2) ? $$ (b) Use a graphing calculator or computer to sketch a slope field for this differential equation. Use the slope field to determine whether each equilibrium solution is stable or unstable.

Show that, for any constant \(P_{0}\), the function \(P=P_{0} e^{t}\) satisfies the equation $$ \frac{d P}{d t}=P $$

(a) Find the equilibrium solution of the equation $$ \frac{d y}{d t}=0.5 y-250 . $$ (b) Find the general solution of this equation. (c) Graph several solutions with different initial values. (d) Is the equilibrium solution stable or unstable?

Suppose \(Q=C e^{k t}\) satisfies the differential equation $$ \frac{d Q}{d t}=-0.03 Q $$ What (if anything) does this tell you about the values of \(C\) and \(k\) ?

Is there a value of \(n\) which makes \(y=x^{n}\) a solution to the equation \(13 x(d y / d x)=y ?\) If so, what value?

A yam is put in a \(200^{\circ} \mathrm{C}\) oven and heats up according to the differential equation \(\frac{d H}{d t}=-k(H-200), \quad\) for \(k\) a positive constant. (a) If the yam is at \(20^{\circ} \mathrm{C}\) when it is put in the oven, solve the differential equation. (b) Find \(k\) using the fact that after 30 minutes the temperature of the yam is \(120^{\circ} \mathrm{C}\).

Find the values of \(k\) for which \(y=x^{2}+k\) is a solution to the differential equation \(2 y-x y^{\prime}=10\).

At \(1: 00 \mathrm{pm}\) one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is \(68^{\circ} \mathrm{F}\) in your house. At \(10: 00 \mathrm{pm}\), it is \(57^{\circ} \mathrm{F}\) in the house, and you notice that it is \(10^{\circ} \mathrm{F}\) outside. (a) Assuming that the temperature, \(T\), in your home obeys Newton's Law of Cooling, write the differential equation satisfied by \(T\). (b) Solve the differential equation to estimate the temperature in the house when you get up at \(7: 00 \mathrm{am}\) the next morning. Should you worry about your water pipes freezing? (c) What assumption did you make in part (a) about the temperature outside? Given this (probably incorrect) assumption, would you revise your estimate up or down? Why?

Match solutions and differential equations. (Note: Each equation may have more than one solution, or no solution.) (a) \(\frac{d y}{d x}=\frac{y}{x}\) (I) \(y=x^{3}\) (b) \(\frac{d y}{d x}=3 \frac{y}{x}\) (II) \(y=3 x\) (c) \(\frac{d y}{d x}=3 x\) (III) \(y=e^{3 x}\) (d) \(\frac{d y}{d x}=y\) (IV) \(y=3 e^{x}\) (e) \(\frac{d y}{d x}=3 y\) (V) \(y=x\)

A drug is administered intravenously at a constant rate of \(r\) mg/hour and is excreted at a rate proportional to the quantity present, with constant of proportionality \(\alpha>0 .\) (a) Solve a differential equation for the quantity, \(Q\), in milligrams, of the drug in the body at time \(t\) hours. Assume there is no drug in the body initially. Your answer will contain \(r\) and \(\alpha\). Graph \(Q\) against \(t\). What is \(Q_{\infty}\), the limiting long-run value of \(Q\) ? (b) What effect does doubling \(r\) have on \(Q_{\infty}\) ? What effect does doubling \(r\) have on the time to reach half the limiting value, \(\frac{1}{2} Q_{\infty}\) ? (c) What effect does doubling \(\alpha\) have on \(Q_{\infty}\) ? On the time to reach \(\frac{1}{2} Q_{\infty}\) ?

Some people write the solution of the initial value problem $$ \frac{d y}{d t}=k(y-A) \quad y=y_{0} \text { at } t=0 $$ in the form $$ \frac{y-A}{y_{0}-A}=e^{k t} $$ Show that this formula gives the correct solution for \(y\), assuming \(y_{0} \neq A\).