Chapter 7

In Exercises 1 and 2 you are given the Lotka-Volterra equations describing the relationship between the prey population (in hundreds) at time \(t, x(t)\), and the predator population (in tens) at time \(t, y(t) .\) (a) Find the equilibrium points of the system. (b) Find an expression for \(d y / d x\) and use it to draw a direction field for the resulting differential equation in the xy-plane. (c) Sketch some solution curves for the differential equation found in part (b). $$ \begin{array}{l} \frac{d x}{d t}=2.4 x-1.2 x y \\ \frac{d y}{d t}=-y+0.8 x y \end{array} $$

A logistic differential equation describing population growth is given. Use the equation to find (a) the growth constant and (b) the carrying capacity of the environment. \(\frac{d P}{d t}=0.02 P\left(1-\frac{P}{1000}\right)\)

Show that \(y=\frac{1}{2}+\frac{3}{x^{2}}\) is a solution of the differential equation \(x y^{\prime}+2 y=1\) on any interval that does not contain \(x=0 .\)

Determine whether the differential equation is linear. $$ x^{2} y^{\prime}+e^{x} y=4 $$

You are given the Lotka-Volterra equations describing the relationship between the prey population (in hundreds) at time \(t, x(t)\), and the predator population (in tens) at time \(t, y(t) .\) (a) Find the equilibrium points of the system. (b) Find an expression for \(d y / d x\) and use it to draw a direction field for the resulting differential equation in the xy-plane. (c) Sketch some solution curves for the differential equation found in part (b). $$ \begin{array}{l} \frac{d x}{d t}=5 x-2 x y \\ \frac{d y}{d t}=-0.6 y+0.2 x y \end{array} $$

A logistic differential equation describing population growth is given. Use the equation to find (a) the growth constant and (b) the carrying capacity of the environment. \(\frac{d P}{d t}=0.03 P-0.000006 P^{2}\)

Show that \(y=C e^{-2 x}+e^{x}\) is a solution of the differential equation \(y^{\prime}+2 y=3 e^{x}\) on \((-\infty, \infty)\).

In Exercises 3 and 4 you are given the phase curve associated with a system of predator-prey equations, where \(x(t)\) denotes the prey (caribou) population, in hundreds, and \(y(t)\) denotes the predator (wolves) population, in tens, at time t. (a) Describe how each population changes over time t starting from \(t=0 .\) (b) Make a rough sketch of the graphs of \(x\) and \(y\) as a function of \(t\) on the same set of axes.

Determine whether the differential equation is linear. $$ y \cos y+\frac{1}{x} \frac{d y}{d x}-\ln x=0 $$

A logistic differential equation describing population growth is given. Use the equation to find (a) the growth constant and (b) the carrying capacity of the environment. \(\frac{d P}{d t}=P\left(0.5-\frac{P}{1000}\right)\)

Show that \(y=x^{4}+3 x^{3}\) is a solution of the initial-value problem \(x y^{\prime}-3 y=x^{4}, y(1)=4\) on \((-\infty, \infty)\).

You are given the phase curve associated with a system of predator-prey equations, where \(x(t)\) denotes the prey (caribou) population, in hundreds, and \(y(t)\) denotes the predator (wolves) population, in tens, at time t. (a) Describe how each population changes over time t starting from \(t=0 .\) (b) Make a rough sketch of the graphs of \(x\) and \(y\) as a function of \(t\) on the same set of axes.

Determine whether the differential equation is linear. $$ y^{2} \frac{d x}{d y}+3 x=\tan y $$

Show that \(y=\sin x-\cos x\) is a solution of the initial-value problem \(\cos x \frac{d y}{d x}+y \sin x=1, y(0)=-1\) on the interval \((-\infty, \infty) .\)

In Exercises \(5-16\), solve the differential equation. $$ \frac{d y}{d x}+2 y=e^{2 x} $$

Solve the differential equation. $$ x \frac{d y}{d x}+3 y=2 $$

Suppose that a solution of the second-order differential equation \(y^{\prime \prime}-y^{\prime}-2 y=0\) has the form \(y=e^{m x}\). a. Find an equation that \(m\) must satisfy. b. Solve the equation found in part (a). c. Write two solutions of the differential equation. d. Verify the results of part (c) directly.

Solve the differential equation. $$ x y^{\prime}+y=x^{3} $$

Consider the predator-prey equations $$ \begin{array}{l} \frac{d x}{d t}=a x-b x y \\ \frac{d y}{d t}=-r y+s x y \end{array} $$ where \(a, b, r\), and \(s\) are positive constants.

Use the given logistic equation to find (a) the growth constant, (b) the carrying capacity of the environment, and (c) the initial population. $$ P(t)=\frac{8000}{2+798 e^{-0.02 t}} $$

In Exercises 7 and 8, the general solution of a differential equation is given. (a) Find the particular solution that satisfies the given initial condition. (b) Plot the solution curves correspond. ing to the given values of \(C\). Indicate the solution curve that corresponds to the solution found in part (a). $$ \begin{array}{l} x \frac{d y}{d x}+y=x^{3}, \quad y=\frac{C}{x}+\frac{x^{3}}{4} ; \quad y(1)=\frac{5}{4} \\ C=-2,-1,0,1,2 \end{array} $$

In nature, the population of aphids (small insects that suck plant juices) is held in check by ladybugs. Assume that the population of aphids (in thousands), \(x(t)\), and the population of ladybugs (in hundreds), \(y(t)\), satisfy the equations $$ \begin{array}{l} \frac{d x}{d t}=2 x-1.2 x y \\ \frac{d y}{d t}=-y+0.8 x y \end{array} $$ a, Find the equilibrium points and interpret your results. b. A direction field for the differential equation $$ \frac{d y}{d x}=\frac{-y+0.8 x y}{2 x-1.2 x y} $$ is shown. Sketch a phase portrait superimposed over the direction field.

Solve the differential equation. $$ y \sin x+y^{\prime} \cos x=1 $$

Use the given logistic equation to find (a) the growth constant, (b) the carrying capacity of the environment, and (c) the initial population. $$ P(t)=\frac{100 e^{0.2 t}}{e^{0.2 t}+19} $$

In Exercises 7 and 8, the general solution of a differential equation is given. (a) Find the particular solution that satisfies the given initial condition. (b) Plot the solution curves correspond. ing to the given values of \(C\). Indicate the solution curve that corresponds to the solution found in part (a). $$ \begin{array}{l} y \frac{d y}{d x}-e^{2 x}=0, \quad y^{2}=e^{2 x}+C ; \quad y(0)=1 ; \\ C=-2,-1,0,1,2 \end{array} $$

In the Lotka-Volterra model it was assumed that an unlimited amount of food was available to the prey. In a situation in which there is a finite amount of natural resources available to the prey, the Lotka-Volterra model can be modified to reflect this situation. Consider the following system of differential equations: $$ \begin{array}{l} \frac{d x}{d t}=k x\left(1-\frac{x}{L}\right)-a x y \\ \frac{d y}{d t}=-b y+c x y \end{array} $$ where \(x(t)\) and \(y(t)\) represent the populations of prey and predators, respectively, and \(a, b, c, k\), and \(L\) are positive constants. a. Describe what happens to the prey population in the absence of predators. b. Describe what happens to the predator population in the absence of prey. c. Find all the equilibrium points and explain their significance.

Solve the differential equation. $$ \frac{d y}{d x}-\frac{2 y}{x}=x^{2} \cos 3 x $$

Consider the logistic differential equation \(\frac{d P}{d t}=k P\left(1-\frac{P}{L}\right)\). a. Show that \(P(t)\) grows most rapidly when \(P=L / 2\). b. Show that \(P(t)\) grows most rapidly at time $$ T=\frac{\ln \left(\frac{L}{P_{0}}-1\right)}{k} $$ where \(P_{0}\) is the initial population.

Use a computer algebra system \((C A S)\) to draw a direction field for the differential equation. Then sketch approximate solution curves passing through the given points by hand superimposed over the direction field. Compare your sketch with the solution curve obtained by using a CAS. \(y^{\prime}=y\) a. \((0,-1)\) b. \((0,0)\) c. \((0,1)\)

In Exercises \(9-18\), solve the differential equation. $$ \frac{d y}{d x}=3 x y-2 x, \quad y(0)=1 $$

Solve the differential equation. $$ \frac{d y}{d x}+y \cot x=\cos x $$

During a flu epidemic the number of children in the Woodbridge Community School System who contracted influenza after \(t\) days was given by $$ Q(t)=\frac{1000}{1+199 e^{-0.8 t}} $$ a. How many children were stricken by the flu after the first day? b. How many children had the flu after 10 days? c. How many children eventually contracted the disease?

Use a computer algebra system \((C A S)\) to draw a direction field for the differential equation. Then sketch approximate solution curves passing through the given points by hand superimposed over the direction field. Compare your sketch with the solution curve obtained by using a CAS. \(y^{\prime}=y-2\) a. \((0,1)\) b. \((0,2)\) c. \((0,4)\)

In Exercises \(9-18\), solve the differential equation. $$ \frac{d y}{d x}=\frac{x+1}{y^{2}} $$

Consider the system of equations $$ \begin{array}{l} \frac{d x}{d t}=k_{1} x\left(1-\frac{x}{L_{1}}\right)-a x y \\ \frac{d y}{d t}=k_{2} y\left(1-\frac{y}{L_{2}}\right)-b x y \end{array} $$ where \(x(t)\) and \(y(t)\) give the populations of two species \(A\) and \(B\), respectively, and \(k_{1}, k_{2}, L_{1}, L_{2}, a\), and \(b\) are positive constants. a. Describe what happens to the population of \(A\) in the absence of \(B\). b. Describe what happens to the population of \(B\) in the absence of \(A\). c. Give a physical interpretation of the roles played by the terms \(a x y\) and \(b x y\), and explain why the equations are called competing species equations. (Examples of competing species are trout and bass.) d. Find the equilibrium points and interpret your results.

Solve the differential equation. $$ x y^{\prime}-y=2 x(\ln x)^{2} $$

The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders. The percentage of teachers who are lay teachers is given by $$ f(t)=\frac{98}{1+2.77 e^{-t}} \quad 0 \leq t \leq 4 $$ where \(t\) is measured in decades, with \(t=0\) corresponding to the beginning of 1960 . a. What percentage of teachers were lay teachers at the beginning of 1990 ? b. Find the year when the percentage of lay teachers was increasing most rapidly.

Use a computer algebra system \((C A S)\) to draw a direction field for the differential equation. Then sketch approximate solution curves passing through the given points by hand superimposed over the direction field. Compare your sketch with the solution curve obtained by using a CAS. \(y^{\prime}=x+y+1\) a. \((0,-2)\) b. \((0,0)\) c. \((0,1)\)

In Exercises \(9-18\), solve the differential equation. $$ \frac{d y}{d x}=x^{2} y $$

A model for the populations of trout, \(x\), and bass, \(y\), that compete for food and space is given by $$ \begin{array}{l} \frac{d x}{d t}=0.6 x\left(1-\frac{x}{4}\right)-0.01 x y \\ \frac{d y}{d t}=0.1 y\left(1-\frac{y}{2}\right)-0.01 x y \end{array} $$ where \(x\) and \(y\) are in thousands. a. Find the equilibrium points of the system. b. Plot the direction field for \(d y / d x\). c. Plot the phase curve that satisfies the initial condition \((6,6)\) superimposed upon the direction field found in part (b). Does this agree with the result of part (a)? d. Interpret your result.

Solve the differential equation. $$ \begin{array}{l} \left(\cos y-x e^{y}\right) d y=e^{y} d x\\\ \text { Hint: Consider } x=f(y) \text { . } \end{array} $$

On the basis of data compiled by the World Health Organization, it is estimated that the number of people living with HIV worldwide from 1985 through 2006 is $$ N(t)=\frac{39.88}{1+18.94 e^{-0.2957 t}} \quad 0 \leq t \leq 21 $$ where \(N(t)\) is measured in millions and \(t\) in years with \(t=0\) corresponding to the beginning of 1985 . a. How many people were living with HIV worldwide at the beginning of \(1985 ?\) At the beginning of \(2005 ?\) b. Assuming that the trend continued, how many people were living with HIV worldwide at the beginning of \(2008 ?\)

Use a computer algebra system \((C A S)\) to draw a direction field for the differential equation. Then sketch approximate solution curves passing through the given points by hand superimposed over the direction field. Compare your sketch with the solution curve obtained by using a CAS. \(y^{\prime}=\frac{1}{4} x^{2}+y\) a. \((0,-2)\) b. \((0,1)\) c. \((1,3)\)

In Exercises \(9-18\), solve the differential equation. $$ \frac{d y}{d x}=-\frac{x y}{x+1} $$

Solve the differential equation. $$ (t+1) \frac{d y}{d t}+y=t, \quad t>-1 $$

Use a computer algebra system \((C A S)\) to draw a direction field for the differential equation. Then sketch approximate solution curves passing through the given points by hand superimposed over the direction field. Compare your sketch with the solution curve obtained by using a CAS. \(y^{\prime}=-\frac{x}{y}\) a. \((-1,1)\) b. \((2,0)\) c. \((0,4)\)

In Exercises \(9-18\), solve the differential equation. $$ y^{\prime}=\frac{2 y+3}{x^{2}} $$

Solve the differential equation. $$ x y^{\prime}+(1+x) y=e^{-x}(1+\cos 2 x) $$

The rate of growth of a certain type of plant is described by a logistic differential equation. Botanists have estimated the maximum theoretical height of such plants to be 30 in. At the beginning of an experiment, the height of a plant was 5 in., and the plant grew to 12 in. after 20 days. a. Find an expression for the height of the plant after \(t\) days. b. What was the height of the plant after 30 days? c. How long did it take for the plant to reach \(80 \%\) of its maximum theoretical height?

Use a computer algebra system \((C A S)\) to draw a direction field for the differential equation. Then sketch approximate solution curves passing through the given points by hand superimposed over the direction field. Compare your sketch with the solution curve obtained by using a CAS. \(y^{\prime}=x(2-y)\) a. \((0,-1)\) b. \((0,2)\) c. \((0,4)\)