Chapter 31

Let \(x=x(t)\) be the number of hundreds of animals of species \(A\) at time \(t\). Let \(y=y(t)\) be the number of hundreds of animals of species \(B\) at time \(t\). For each system of differential equations, describe the nature of the interaction between the two species. What happens to each species in the absence of the other? (a) \(\left\\{\begin{array}{l}\frac{d x}{d t}=0.02 x-0.001 x^{2}-0.002 x y \\\ \frac{d y}{d t}=0.008 y-0.004 y^{2}-0.001 x y\end{array}\right.\) (b) \(\left\\{\begin{array}{l}\frac{d x}{d t}=0.02 x-0.01 x y \\ \frac{d y}{d t}=-0.01 y+0.08 x y\end{array}\right.\) (c) \(\left\\{\begin{array}{l}\frac{d x}{d t}=0.02 x-0.001 x^{2}+0.002 x y \\\ \frac{d y}{d t}=0.03 y-0.006 y^{2}+0.001 x y\end{array}\right.\)

Do a qualitative analysis of the family of solutions to each of the differential equations below. Then, in another color pen or pencil, highlight the graphs of the solutions corresponding to the given initial conditions. (If the solution is asymptotic to a horizontal line, draw and label that line.) \(\begin{array}{llll}\text { (a) } \frac{d y}{d t}=4 y-8 ; & y(0)=0, & y(0)=-1, & y(0)=3\end{array}\) (b) \(\frac{d y}{d t}=y^{2}-4\) \(y(0)=-1, \quad y(0)=-3, \quad y(0)=4\) (c) \(\frac{d y}{d t}=(y-1)(y-2)(y+1) ; \quad y(0)=0, \quad y(0)=3\) \(\begin{array}{llll}\text { (d) } \frac{d y}{d t}=y^{2}+5 y-6 ; & y(0)=-5, & y(0)=-7, & y(0)=2\end{array}\)

Money is deposited in a bank account with a nominal annual interest rate of \(4 \%\) compounded continuously. Let \(M=M(t)\) be the amount of money in the account at time \(t\). (a) Write a differential equation whose solution is \(M(t)\). Assume there are no additional deposits and no withdrawals. (b) Suppose money is being added to the account continuously at a rate of \(\$ 1000\) per year and no withdrawals are made. Write a differential equation whose solution is \(M(t)\)

Solve the differential equations. \(y^{\prime \prime}-4 y^{\prime}=-3\)

Solve the given differential equation. \(\frac{d y}{d x}=\frac{x}{2 y}\)

Which one of the following is a solution to the differential equation \(y^{\prime}(t)=-5 y\) ? (a) \(y=e^{5 t}\) (b) \(y=t^{2}\) (c) \(y=e^{-5 t}\) (d) \(y=\sin (5 t)\)

For each system of differential equations, nd the nullclines and identify the equilibrium solutions. $$ \left\\{\begin{array}{l} \frac{d x}{d t}=0.02 x-0.001 x^{2}-0.002 x y \\ \frac{d y}{d x}=0.008 y-0.004 y^{2}-0.001 x y \end{array}\right. $$

We can construct a model for the spread of a disease by assuming that people are being infected at a rate proportional to the product of the number of people who have already been infected and the number of those who have not. Let \(P(t)\) denote the number of infected people at time \(t\) and \(N\) denote the total population affected by the epidemic. Assume \(N\) is xed throughout the time period we are considering. We are assuming that every member of the population is susceptible to the disease and the disease is long in duration (there are no recoveries during the time period we are analyzing) but not fatal (no deaths during this period). The assumption that people are being infected at a rate proportional to the product of those who are infected and those who are not could re ect a contagious disease where the sick are not isolated. Write a differential equation whose solution is \(P(t)\).

Solve the differential equations. \(y^{\prime \prime}-2 y^{\prime}+y=0\)

Solve the given differential equation. \(\frac{d y}{d x}=x^{2} y\)

Which one of the following is a solution to the differential equation \(y^{\prime \prime}(t)=-25 y ?\) (a) \(y=e^{5 t}\) (b) \(y=t^{2}\) (c) \(y=e^{-5 t}\) (d) \(y=\sin (5 t)\)

For each system of differential equations, nd the nullclines and identify the equilibrium solutions. $$ \left\\{\begin{array}{l} \frac{d x}{d t}=0.02 x-0.01 x y \\ \frac{d y}{d t}=-0.01 y+0.08 x y \end{array}\right. $$

Solutes in the bloodstream enter cells through osmosis, the diffusion of uid through a semipermeable membrane until the concentration of uid on both sides of the membrane is equal. Suppose that the concentration of a certain solute in the bloodstream is maintained at a constant level of \(K \mathrm{mg} /\) cubic \(\mathrm{cm}\). Let s consider \(f(t)\), the concentration of the solute inside a certain cell at time \(t .\) The rate at which the concentration of the solute inside the cell is changing is proportional to the difference between the concentration of the solute in the bloodstream and its concentration inside the cell. (a) Set up the differential equation whose solution is \(y=f(t)\). (b) Sketch a solution assuming that \(f(0)

Solve the differential equations. \(y^{\prime \prime}+25 y=0\)

Solve the given differential equation. \(\frac{d y}{d x}=x y^{2}\)

(a) Verify that \(y(t)=C e^{k t}\) is a solution to the differential equation \(\frac{d y}{d t}=k y\). (b) Verify that \(y=k e^{t}\) is not a solution to \(\frac{d y}{d t}=k y\). (c) Verify that \(y=e^{k t}+C\) is not a solution to \(\frac{d y}{d t}=k y\).

For each system of differential equations, nd the nullclines and identify the equilibrium solutions. $$ \left\\{\begin{array}{l} \frac{d x}{d t}=0.02 x-0.001 x^{2}+0.002 x y \\ \frac{d y}{d t}=0.03 y-0.006 y^{2}+0.001 x y \end{array}\right. $$

Consider the differential equation \(\frac{d P}{d t}=k P(L-P)\), where \(k\) and \(L\) are positive constants. (a) For what values of \(P\) is \(\frac{d P}{d t}\) zero? (b) Show that \(P(t)=\frac{L}{1+C e^{-k L t}}\), where \(C\) is a constant, is a solution of the logistic equation above.

A large garbage dump sits on the outskirts of Cairo. Garbage is being deposited at the dump at a rate of \(T\) tons per month. Scavengers and salvagers frequent the dump and haul off refuse from the site. The rate at which garbage is being hauled off is proportional to the tonnage at the site. Let \(G(t)\) be the number of tons of garbage in the dump. Write a differential equation whose solution is \(G(t) .\) The basic framework is rate of change of \(G=\) rate of increase of \(G-\) rate of decrease of \(G\).

Solve the differential equations. \(y^{\prime \prime}+5 y=0\)

Solve the given differential equation. \(\frac{d y}{d x}=\frac{y}{x}\)

Let \(x=x(t)\) be the number of thousands of animals of species \(A\) at time \(t\). Let \(y=y(t)\) be the number of thousands of animals of species \(B\) at time \(t\). Suppose \(\left\\{\begin{array}{l}\frac{d x}{d t}=x-0.5 x y \\ \frac{d y}{d t}=y-0.5 x y .\end{array}\right.\) (a) Is the interaction between species \(A\) and \(B\) symbiotic, competitive, or a predatorprey relationship? (b) What are the equilibrium populations? (c) Find the nullclines and draw directed horizontal and vertical tangent lines in the phase-plane (as in Figures \(31.28\) and 31.30). (d) The nullclines divide the first quadrant of the phase-plane into four regions. In each region determine the general direction of the trajectories. (e) If \(x=0\), what happens to \(y(t) ?\) How is this indicated in the phase- plane? If \(y=0\), what happens to \(x(t) ?\) How is this indicated in the phase- plane? (f) Use the information gathered in parts (b) through (e) to sketch representative solution trajectories in the phase-plane. Include arrows indicating the direction the trajectories are traveled. (g) For each of the initial conditions given below, describe how the number of species of \(A\) and \(B\) change with time and what the situation will look like in the long run. i. \(x(0)=2 \quad y(0)=1.8\) ii. \(x(0)=2 \quad y(0)=2.3\) iii. \(x(0)=2.2 \quad y(0)=2\) (h) Does this particular model support or challenge Charles Darwin's principle of competitive exclusion?

The population of a town in the south of Bangladesh has been growing exponentially. However, recent flooding has alarmed residents and people are leaving the town at a rate of \(N\) thousand people per year, where \(N\) is a constant. The rate of change of the population of the town can be modeled by the differential equation $$ \frac{d P}{d t}=0.02 P-N $$ where \(P=P(t)\) is the number of people in the town in thousands. (a) If \(P(0)=100\), what is the largest yearly exodus rate the town can support in the long run? (b) How big must the population of the town be in order to support the loss of 1000 people per year?

Elmer takes out a $$\$ 100,000$$ loan for a house. He pays money back at a rate of $$\$ 12,000$$ per year. The bank charges him interest at a rate of \(8.5 \%\) per year compounded continuously. Make a continuous model of his economic situation. Write a differential equation whose solution is \(B(t)\), the balance he owes the bank at time \(t\).

Solve the differential equations. \(y^{\prime \prime}+5 y^{\prime}=0\)

Solve the given differential equation. \(\frac{d y}{d x}=\frac{x-1}{2 y+1}\)

Determine which of the following functions are solutions to each of the differential equations below. (A given differential equation may have more than one solution.) Differential Equations: i. \(\frac{d y}{d t}=t\) ii. \(\frac{d y}{d t}=y\) iii. \(\frac{d y}{d t}=e^{t}\) iv. \(\frac{d^{2} y}{d t^{2}}=4 y\) Solution choices: (a) \(y=\frac{t^{2}}{2}\) (b) \(y=\frac{t^{2}}{2}+5\) (c) \(y=e^{-2 t}\) (d) \(y=e^{t}+5\) (e) \(y=2 e^{t}\) (f) \(y=e^{2 t}\) (g) \(y=5 e^{2 t}\) (h) \(y=e^{2 I}+5\)

Give systems of differential equations modeling competition between two species. In each problem nd the nullclines. The nullclines will divide the phase-plane into regions; nd the direction of the trajectories in each region. Use this information to sketch a phase-plane portrait. Then interpret the implications of your portrait for the long-term outcome of the competition. \(x(t)\) and \(y(t)\) give the number of thousands of animals of species \(A\) and \(B\), respectively. $$ \left\\{\begin{array}{l} \frac{d x}{d t}=0.03 x-0.01 x^{2}-0.01 x y \\ \frac{d y}{d t}=0.05 y-0.01 y^{2}-0.01 x y \end{array}\right. $$

Let s suppose that the population in a certain country has a growth rate of \(2 \%\) and a population of 9 million at a time we ll designate as \(t=0 .\) Due to the political and economic situation, there is a massive rearrangement of populations in the region. The immigration and emigration rates are both constant, with people entering the country at a rate of 100,000 per year and leaving at a rate of 300,000 per year. Let \(P=P(t)\) be the population in millions at time \(t\). (a) Write a differential equation re ecting the situation. Keep in mind that \(P\) is in millions. (b) If this situation goes on inde nitely, what will happen to the country s population? (c) What initial population would support a net emigration of 200,000 per year?

Solve the differential equations. \(y^{\prime \prime}=y^{\prime}+2 y\)

Solve the given differential equation. \(2 y^{\prime}-y=1\)

Which of the following is a solution to the differential equation $$ y^{\prime \prime}-y^{\prime}-6 y=0 ? $$ (a) \(y=C e^{t}\) (b) \(y=\sin 2 t\) (c) \(y=5 e^{3 t}+e^{-2 r}\) (d) \(y=e^{3 t}-2\)

Give systems of differential equations modeling competition between two species. In each problem nd the nullclines. The nullclines will divide the phase-plane into regions; nd the direction of the trajectories in each region. Use this information to sketch a phase-plane portrait. Then interpret the implications of your portrait for the long-term outcome of the competition. \(x(t)\) and \(y(t)\) give the number of thousands of animals of species \(A\) and \(B\), respectively. $$ \left\\{\begin{array}{l} \frac{d x}{d t}=0.04 x-0.02 x^{2}-0.01 x y \\ \frac{d y}{d t}=0.04 y-0.01 y^{2}-0.01 x y \end{array}\right. $$

In the beginning of a chemical reaction there are 600 moles of substance \(\mathrm{A}\) and none of substance \(\mathrm{B}\). Over the course of the reaction, the 600 moles of substance \(\mathrm{A}\) are converted to 600 moles of substance B. (Each molecule of A is converted to a molecule of \(\mathrm{B}\) via the reaction.) Suppose the rate at which \(\mathrm{A}\) is turning into \(\mathrm{B}\) is proportional to the product of the number of moles of \(\mathrm{A}\) and the number of moles of \(\mathrm{B}\). (a) Let \(N=N(t)\) be the number of moles of substance \(\mathrm{A}\) at time \(t .\) Translate the statement above into mathematical language. (Note: The number of moles of substance B should be expressed in terms of the number of moles of substance A.) (b) Using your answer to part (a), nd \(\frac{d^{2} N}{d t^{2}}\). Your answer will involve the proportionality constant used in part (a). (c) \(N(t)\) is a decreasing function. The rate at which \(N\) is changing is a function of \(N\), the number of moles of substance A. When the rate at which \(\mathrm{A}\) is being converted to \(\mathrm{B}\) is highest, how many moles are there of substance \(\mathrm{A}\) ?

Solve the differential equations. \(3 y^{\prime \prime}+3 y^{\prime}+3 y=0\)

Solve the given differential equation. \(\frac{d y}{d x}-y^{2}=1\)

Which of the following is a solution to the differential equation $$ y^{\prime \prime}+9 y=0 ? $$ (a) \(y=e^{3 t}+e^{-3 t}\) (b) \(y=C e^{t}-t\) (c) \(y=C\left(t^{2}+t\right)\) (d) \(y=\sin 3 t+6\) (e) \(y=5 \cos 3 t\)

Give systems of differential equations modeling competition between two species. In each problem nd the nullclines. The nullclines will divide the phase-plane into regions; nd the direction of the trajectories in each region. Use this information to sketch a phase-plane portrait. Then interpret the implications of your portrait for the long-term outcome of the competition. \(x(t)\) and \(y(t)\) give the number of thousands of animals of species \(A\) and \(B\), respectively. $$ \left\\{\begin{array}{l} \frac{d x}{d t}=0.01 x(2-2 x-y) \\ \frac{d y}{d t}=0.01 y(1-y-0.25 x) \end{array}\right. $$

Essay question: One of your classmates is puzzled by what it means to solve a differential equation. He has two questions for you. Answer them in plain English. (a) How can you tell if something is a solution to a differential equation? (b) What is the difference between a general and a particular solution? Is the difference always that the general solution just has a "plus \(C\) " at the end? Are there some cases where that's the only difference?

There are many places in the world where populations are changing and immigration and/or emigration play a big role. People may move to nd food, or to nd jobs, or to ee political or religious persecution. Pick a situation that interests you. You could look at the number of Tibetans in Tibet, or the number of Tibetans in India, or the number of lions in the Serengeti, or the number of tourists in Nepal. Get some data and try to model the population dynamics using a differential equation. What simplifying assumptions have you made?

Solve the given differential equation. \(\frac{d y}{d t}=t \cos ^{2} y\)

Which of the following is a solution to the differential equation $$ \frac{d y}{d t}=y+1 ? $$ (a) \(y=C e^{t}\) (b) \(y=C e^{t}-t\) (c) \(y=C\left(t^{2}+t\right)\) (d) \(y=C e^{t}-1\) (e) \(y=C e^{-t}+1\)

Suppose we modify the Volterra predator-prey equations to reflect competition among prey for limited resources and competition among predators for limited resources. The equations would be of the form $$ \left\\{\begin{array}{l} \frac{d x}{d t}=k_{1} x-k_{2} x^{2}-k_{3} x y \\ \frac{d y}{d t}=-k_{4} y-k_{5} y^{2}+k_{6} x y \end{array}\right. $$ where \(k_{1}, k_{2}, \ldots, k_{6}\) are positive constants. Consider the system $$ \left\\{\begin{array}{l} \frac{d x}{d t}=x(1-0.5 x-y) \\ \frac{d y}{d t}=y(-1-0.5 y+x) \end{array}\right. $$ (a) Find the equilibrium points. (b) Do a qualitative phase-plane analysis. (In fact, solution trajectories will spiral in toward the non-trivial equilibrium point.)

Let \(P(t)\) be the number of crocodiles in a mud hole at time \(t\). Suppose \(\frac{d P}{d t}=0.01 P-0.0025 P^{2}\) (a) What is the carrying capacity of the mud hole? (b) Find \(\frac{d^{2} P}{d t^{2}}\). Remember: You are differentiating with respect to \(t\), so the derivative of \(P\) is not 1 . (c) Use your answer to part (b) to determine how many crocodiles are in the mud hole when the number of crocodiles is increasing most rapidly. (d) Sketch a solution curve if the number of crocodiles in the mud hole at time \(t=0\) is 3. (Label the vertical axis. You need not calibrate the \(t\) -axis.)

Find the particular solution corresponding to the initial conditions given. \(\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x, \quad x(0)=-1, \quad x^{\prime}(0)=0\)

Solve the given differential equation. \(2 \frac{d y}{d x}-3 x y=0\)

Is \(y=\frac{x e^{x}}{2}+\frac{e^{x}}{3 x}\) a solution to the differential equation \(x \frac{d y}{d x}+(1-x) y=x e^{x} ?\) Justify your answer.

Sketch a representative family of solutions to the following differential equations. You need not take a second derivative. (a) \(\frac{d y}{d t}=y\left(y^{2}-4\right)\) (b) \(\frac{d y}{d t}=y^{2}(y-2)\)

Find the particular solution corresponding to the initial conditions given. \(2 x^{\prime \prime}+6 x=0, \quad x(0)=0, \quad x^{\prime}(0)=4\)

Solve the given differential equation. \(\frac{d y}{d x}=\frac{\cos x}{-\sin y}\)