Chapter 9

Solve the differential equations in Exercises \(1-14\) $$x \frac{d y}{d x}+y=e^{x}, \quad x>0$$

In Exercises \(1-8\) \begin{equation}\begin{array}{l}{\text { a. Identify the equilibrium values. Which are stable and which }} \\ {\text { are unstable? }} \\ {\text { b. Construct a phase line. Identify the signs of } y^{\prime} \text { and } y^{\prime \prime} \text { . }} \\ {\text { c. Sketch several solution curves. }}\end{array}\end{equation} $$\frac{d y}{d x}=(y+2)(y-3)$$

For the system \((2 a)\) and \((2 b),\) show that any trajectory starting on the unit circle \(x^{2}+y^{2}=1\) will traverse the unit circle in a periodic solution. First introduce polar coordinates and rewrite the system as \(d r / d t=r\left(1-r^{2}\right)\) and \(-d \theta / d t=-1\)

In Exercises \(1-8\) \begin{equation}\begin{array}{l}{\text { a. Identify the equilibrium values. Which are stable and which }} \\ {\text { are unstable? }} \\ {\text { b. Construct a phase line. Identify the signs of } y^{\prime} \text { and } y^{\prime \prime} \text { . }} \\ {\text { c. Sketch several solution curves. }}\end{array}\end{equation} $$\frac{d y}{d x}=y^{2}-4$$

Solve the differential equations in Exercises \(1-14\) $$e^{x} \frac{d y}{d x}+2 e^{x} y=1$$

In Exercises \(1-8\) \begin{equation}\begin{array}{l}{\text { a. Identify the equilibrium values. Which are stable and which }} \\ {\text { are unstable? }} \\ {\text { b. Construct a phase line. Identify the signs of } y^{\prime} \text { and } y^{\prime \prime} \text { . }} \\ {\text { c. Sketch several solution curves. }}\end{array}\end{equation} $$\frac{d y}{d x}=y^{3}-y$$

Solve the differential equations in Exercises \(1-14\) $$x y^{\prime}+3 y=\frac{\sin x}{x^{2}}, \quad x>0$$

In Exercises \(1-8\) \begin{equation}\begin{array}{l}{\text { a. Identify the equilibrium values. Which are stable and which }} \\ {\text { are unstable? }} \\ {\text { b. Construct a phase line. Identify the signs of } y^{\prime} \text { and } y^{\prime \prime} \text { . }} \\ {\text { c. Sketch several solution curves. }}\end{array}\end{equation} $$\frac{d y}{d x}=y^{2}-2 y$$

Solve the differential equations in Exercises \(1-14\) $$y^{\prime}+(\tan x) y=\cos ^{2} x, \quad-\pi / 2 < x < \pi / 2$$

Consider another competitive-hunter model defined by $$ \begin{aligned} \frac{d x}{d t} &=a\left(1-\frac{x}{k_{1}}\right) x-b x y \\\ \frac{d y}{d t} &=m\left(1-\frac{y}{k_{2}}\right) y-n x y \end{aligned} $$ where \(x\) and \(y\) represent trout and bass populations, respectively. $$ \begin{array}{l}{\text { a. What assumptions are implicitly being made about the }} \\ {\text { growth of trout and bass in the absence of competition? }} \\\ {\text { b. Interpret the constants } a, b, m, n, k_{1}, \text { and } k_{2} \text { in terms of the }} \\ {\text { physical problem. }}\\\\{\text { c. Perform a graphical analysis: }} \\ {\text { i) Find the possible equilibrium levels. }} \\ {\text { ii) Determine whether coexistence is possible. }} \\ {\text { iii) Pick several typical starting points and sketch typical }} \\ {\text { trajectories in the phase plane. }} \\ {\text { iv) Interpret the outcomes predicted by your graphical }} \\ {\text { analysis in terms of the constants } a, b, m, n, k_{1}, \text { and } k_{2} \text { . }}\end{array} $$ Note: When you get to part (ii), you should realize that five cases exist. You will need to analyze all five cases.

In Exercises \(1-8\) \begin{equation}\begin{array}{l}{\text { a. Identify the equilibrium values. Which are stable and which }} \\ {\text { are unstable? }} \\ {\text { b. Construct a phase line. Identify the signs of } y^{\prime} \text { and } y^{\prime \prime} \text { . }} \\ {\text { c. Sketch several solution curves. }}\end{array}\end{equation} $$y^{\prime}=\sqrt{y}, \quad y>0$$

Solve the differential equations in Exercises \(1-14\) $$x \frac{d y}{d x}+2 y=1-\frac{1}{x}, \quad x>0$$

In Exercises \(5 - 10 ,\) find the orthogonal trajectories of the family of curves. Sketch several members of each family. $$ y = c x ^ { 2 } $$

An economic model Consider the following economic model. Let \(P\) be the price of a single item on the market. Let \(Q\) be the quantity of the item available on the market. Both \(P\) and \(Q\) are functions of time. If one considers price and quantity as two inter- acting species, the following model might be proposed: $$ \begin{aligned} \frac{d P}{d t} &=a P\left(\frac{b}{Q}-P\right) \\ \frac{d Q}{d t} &=c Q(f P-Q) \end{aligned} $$ where \(a, b, c,\) and \(f\) are positive constants. Justify and discuss the adequacy of the model. $$ \begin{array}{l}{\text { a. If } a=1, b=20,000, c=1, \text { and } f=30 \text { , find the equilibrium }} \\ {\text { points of this system. If possible, classify each equilibrium }} \\ {\text { point with respect to its stability. If a point cannot be }} \\ {\text { readily classified, give some explanation. }}\\\\{\text { b. Perform a graphical stability analysis to determine what will }} \\ {\text { happen to the levels of } P \text { and } Q \text { as time increases. }} \\ {\text { c. Give an economic interpretation of the curves that determine }} \\ {\text { the equilibrium points. }}\end{array} $$

In Exercises \(1-8\) \begin{equation}\begin{array}{l}{\text { a. Identify the equilibrium values. Which are stable and which }} \\ {\text { are unstable? }} \\ {\text { b. Construct a phase line. Identify the signs of } y^{\prime} \text { and } y^{\prime \prime} \text { . }} \\ {\text { c. Sketch several solution curves. }}\end{array}\end{equation} $$y^{\prime}=y-\sqrt{y}, \quad y>0$$

Solve the differential equations in Exercises \(1-14\) $$(1+x) y^{\prime}+y=\sqrt{x}$$

Solve the differential equations in Exercises \(1-14\) $$2 y^{\prime}=e^{x / 2}+y$$

Show that the second-order differential equation \(y^{\prime \prime}=F\left(x, y, y^{\prime}\right)\) can be reduced to a system of two first-order differential equations $$ \begin{array}{l}{\frac{d y}{d x}=z} \\ {\frac{d z}{d x}=F(x, y, z)}\end{array} $$ Can something similar be done to the \(n\) th-order differential equation \(y^{(n)}=F\left(x, y, y^{\prime}, y^{n}, \ldots, y^{(n-1)}\right) ?\)

In Exercises \(1-8\) \begin{equation}\begin{array}{l}{\text { a. Identify the equilibrium values. Which are stable and which }} \\ {\text { are unstable? }} \\ {\text { b. Construct a phase line. Identify the signs of } y^{\prime} \text { and } y^{\prime \prime} \text { . }} \\ {\text { c. Sketch several solution curves. }}\end{array}\end{equation} $$y^{\prime}=y^{3}-y^{2}$$

Solve the differential equations in Exercises \(1-14\) $$e^{2 x} y^{\prime}+2 e^{2 x} y=2 x$$

Write an equivalent first-order differential equation and initial condition for \(y .\) \(y=\int_{1}^{x} \frac{1}{t} d t\)

In 1925 Lotka and Volterra introduced the predator-prey equations, a system of equations that models the populations of two species, one of which preys on the other. Let \(x(t)\) represent the number of rabbits living in a region at time \(t,\) and \(y(t)\) the number of foxes in the same region. As time passes, the number of rabbits increases at a rate proportional to their population, and decreases at a rate proportional to the number of encounters between rabbits and foxes. The foxes, which compete for food, increase in number at a rate proportional to the number of encounters with rabbits but decrease at a rate proportional to the number of foxes. The number of encounters between rabbits and foxes is assumed to be proportional to the product of the two populations. These assumptions lead to the autonomous system $$ \begin{array}{l}{\frac{d x}{d t}=(a-b y) x} \\ {\frac{d y}{d t}=(-c+d x) y}\end{array} $$ where \(a, b, c, d\) are positive constants. The values of these constants vary according to the specific situation being modeled. We can study the nature of the population changes without setting these constants to specific values. What happens to the rabbit population if there are no foxes present?

In Exercises \(5 - 10 ,\) find the orthogonal trajectories of the family of curves. Sketch several members of each family. $$ y = c e ^ { - x } $$

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? $$\frac{d P}{d t}=1-2 P$$

Solve the differential equations in Exercises \(1-14\) $$x y^{\prime}-y=2 x \ln x$$

Write an equivalent first-order differential equation and initial condition for \(y .\) \(y=2-\int_{0}^{x}(1+y(t)) \sin t d t\)

In 1925 Lotka and Volterra introduced the predator-prey equations, a system of equations that models the populations of two species, one of which preys on the other. Let \(x(t)\) represent the number of rabbits living in a region at time \(t,\) and \(y(t)\) the number of foxes in the same region. As time passes, the number of rabbits increases at a rate proportional to their population, and decreases at a rate proportional to the number of encounters between rabbits and foxes. The foxes, which compete for food, increase in number at a rate proportional to the number of encounters with rabbits but decrease at a rate proportional to the number of foxes. The number of encounters between rabbits and foxes is assumed to be proportional to the product of the two populations. These assumptions lead to the autonomous system $$ \begin{array}{l}{\frac{d x}{d t}=(a-b y) x} \\ {\frac{d y}{d t}=(-c+d x) y}\end{array} $$ where \(a, b, c, d\) are positive constants. The values of these constants vary according to the specific situation being modeled. We can study the nature of the population changes without setting these constants to specific values. What happens to the fox population if there are no rabbits present?

In Exercises \(5 - 10 ,\) find the orthogonal trajectories of the family of curves. Sketch several members of each family. $$ y = e ^ { k x } $$

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? $$\frac{d P}{d t}=P(1-2 P)$$

Solve the differential equations in Exercises \(1-14\) $$x \frac{d y}{d x}=\frac{\cos x}{x}-2 y, \quad x>0$$

Write an equivalent first-order differential equation and initial condition for \(y .\) \(y=1+\int_{0}^{x} y(t) d t\)

Show that the curves \(2 x ^ { 2 } + 3 y ^ { 2 } = 5\) and \(y ^ { 2 } = x ^ { 3 }\) are orthogonal.

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? $$\frac{d P}{d t}=2 P(P-3)$$

Solve the differential equations in Exercises \(1-14\) $$(t-1)^{3} \frac{d s}{d t}+4(t-1)^{2} s=t+1, \quad t>1$$

Find the family of solutions of the given differential equation and the family of orthogonal trajectories. Sketch both families. $$ \begin{array} { l } { \text { a. } x d x + y d y = 0 } \\ { \text { b. } x d y - 2 y d x = 0 } \end{array} $$

The autonomous differential equations in Exercises \(9-12\) represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? $$\frac{d P}{d t}=3 P(1-P)\left(P-\frac{1}{2}\right)$$

Solve the differential equations in Exercises \(1-14\) $$(t+1) \frac{d s}{d t}+2 s=3(t+1)+\frac{1}{(t+1)^{2}}, \quad t>-1$$

Catastrophic change in logistic growth Suppose that a healthy population of some species is growing in a limited environment and that the current population \(P_{0}\) is fairly close to the carrying capacity \(M_{0 .}\) You might imagine a population of fish living in a freshwater lake in a wilderness area. Suddenly a catastrophe such as the Mount St. Helens volcanic eruption contaminates the lake and destroys a significant part of the food and oxygen on which the fish depend. The result is a new environment with a carrying capacity \(M_{1}\) considerably less than \(M_{0}\) and, in fact, less than the current population \(P_{0}\) . Starting at some time before the catastrophe, sketch a "before-and-after" curve that shows how the fish population responds to the change in environment.

Solve the differential equations in Exercises \(1-14\) $$\sin \theta \frac{d r}{d \theta}+(\cos \theta) r=\tan \theta, \quad 0<\theta<\pi / 2$$

Mixture problem \(A 200\) -gal tank is half full of distilled water. At time \(t = 0 ,\) a solution containing 0.5 lb \(/\) gal of concentrate enters the tank at the rate of 5 gal \(/ \mathrm { min }\) , and the well-stirred mixture is withdrawn at the rate of 3 gal/min. a. At what time will the tank be full? b. At the time the tank is full, how many pounds of concentrate will it contain?

Controlling a population The fish and game department in a certain state is planning to issue hunting permits to control the deer population (one deer per permit). It is known that if the deer population falls below a certain level \(m,\) the deer will become extinct. It is also known that if the deer population rises above the carrying capacity \(M,\) the population will decrease back to \(M\) through disease and malnutrition. \begin{equation}\begin{array}{l}{\text { a. Discuss the reasonableness of the following model for the }} \\ {\text { growth rate of the deer population as a function of time: }}\end{array}\end{equation} $$\frac{d P}{d t}=r P(M-P)(P-m)$$ \begin{equation}\begin{array}{l}{\text { where } P \text { is the population of the deer and } r \text { is a positive con- }} \\ {\text { stant of proportionality. Include a phase line. }}\end{array}\end{equation} \begin{equation}\begin{array}{l}{\text { c. Show that if } P > M \text { for all } t \text { , then } \lim _{t \rightarrow \infty} P(t)=M .} \\ {\text { d. What happens if } P < m \text { for all } t ?} \\ {\text { e. Discuss the solutions to the differential equation. What are }} \\ {\text { the equilibrium points of the model? Explain the dependence }} \\ {\text { of the steady-state value of } P \text { on the initial values of } P . \text { About }} \\ {\text { how many permits should be issued? }}\end{array}\end{equation}

Solve the differential equations in Exercises \(1-14\) $$\tan \theta \frac{d r}{d \theta}+r=\sin ^{2} \theta, \quad 0<\theta<\pi / 2$$

Skydiving If a body of mass \(m\) falling from rest under the action of gravity encounters an air resistance proportional to the square of velocity, then the body's velocity \(t\) seconds into the fall satisfies the equation $$m \frac{d v}{d t}=m g-k v^{2}, \quad k > 0$$ where \(k\) is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is too short to be affected by changes in the air's density.) \begin{equation}\begin{array}{l}{\text { a. Draw a phase line for the equation. }} \\ {\text { b. Sketch a typical velocity curve. }} \\ {\text { c. For a } 110 \text { -lb skydiver }(m g=110) \text { and with time in seconds }} \\ {\text { and distance in feet, a typical value of } k \text { is } 0.005 . \text { What is the }} \\ {\text { diver's terminal velocity? Repeat for a } 200-\text { -lb skydiver. }}\end{array}\end{equation}

Solve the initial value problems in Exercises \(15-20\) $$\frac{d y}{d t}+2 y=3, \quad y(0)=1$$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. \(y^{\prime}=\frac{2 y}{x}, \quad y(1)=-1, \quad d x=0.5\)

Carbon monoxide pollution An executive conference room of a corporation contains 4500\(\mathrm { ft } ^ { 3 }\) of air initially free of carbon monoxide. Starting at time \(t = 0 ,\) cigarette smoke containing 4\(\%\) carbon monoxide is blown into the room at the rate of 0.3\(\mathrm { ft } ^ { 3 } / \mathrm { min }\) . A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3\(\mathrm { ft } ^ { 3 } / \mathrm { min }\) . Find the time when the concentration of carbon monoxide in the room reaches 0.01\(\% .\)

Resistance proportional to \(\sqrt{v}\) A body of mass \(m\) is projected vertically downward with initial velocity \(v_{0}\) . Assume that the re- sisting force is proportional to the square root of the velocity and find the terminal velocity from a graphical analysis.

Solve the initial value problems in Exercises \(15-20\) $$t \frac{d y}{d t}+2 y=t^{3}, \quad t>0, \quad y(2)=1$$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. \(y^{\prime}=x(1-y), \quad y(1)=0, \quad d x=0.2\)

Sailing A sailboat is running along a straight course with the wind providing a constant forward force of 50 lb. The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to five times the boat's speed, and the initial velocity is 1 \(\mathrm{ft} /\) sec. What is the maximum velocity in feet per second of the boat under this wind?