Chapter 9

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(\frac{d y}{d x}=(y+2)(y-3)\)

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

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(\frac{d y}{d x}=y^{2}-4\)

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

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(\frac{d y}{d x}=y^{3}-y\)

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

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(\frac{d y}{d x}=y^{2}-2 y\)

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. a. What assumptions are implicitly being made about the growth of trout and bass in the absence of competition? b. Interpret the constants \(a, b, m, n, k_{1},\) and \(k_{2}\) in terms of the physical problem. c. Perform a graphical analysis: i) Find the possible equilibrium levels. ii) Determine whether coexistence is possible. iii) Pick several typical starting points and sketch typical trajectories in the phase plane. iv ) Interpret the outcomes predicted by your graphical analysis

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(y^{\prime}=\sqrt{y}, \quad y>0\)

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

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 interacting 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. a. If \(a=1, b=20,000, c=1,\) and \(f=30\) , find the equilibrium points of this system. If possible, classify each equilibrium point with respect to its stability. If a point cannot be readily classified, give some explanation. b. Perform a graphical stability analysis to determine what will happen to the levels of \(P\) and \(Q\) as time increases. c. Give an economic interpretation of the curves that determine the equilibrium points.

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(y^{\prime}=y-\sqrt{y}, \quad y>0\)

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

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

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(y^{\prime}=(y-1)(y-2)(y-3)\)

In Exercises \(5-10,\) find the orthogonal trajectories of the family of curves. Sketch sketch skeveral members of each family. $$k x^{2}+y^{2}=1$$

In Exercises \(7-10\) , write an equivalent first-order differential equation and initial condition for \(y .\) $$ y=-1+\int_{1}^{x}(t-y(t)) d t $$

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 $$\begin{aligned} \frac{d y}{d x} &=z \\ \frac{d z}{d x} &=F(x, y, z) \end{aligned}$$ Can something similar be done to the \(n\) th-order differential equation \(y^{(n)}=F\left(x, y, y^{\prime}, y^{\prime \prime}, \ldots, y^{(n-1)}\right) ?\) Lotka-Volterra Equations for a Predator-Prey Model 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 $$\begin{aligned} \frac{d x}{d t} &=(a-b y) x \\ \frac{d y}{d t} &=(-c+d x) y \end{aligned}$$ 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.

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(y^{\prime}=y^{3}-y^{2}\)

In Exercises \(7-10\) , write an equivalent first-order differential equation and initial condition for \(y .\) $$ y=\int_{1}^{x} \frac{1}{t} d t $$

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

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

The autonomous differential equations 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? \(\frac{d P}{d t}=1-2 P\)

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

In Exercises \(7-10\) , write an equivalent first-order differential equation and initial condition for \(y .\) $$ y=2-\int_{0}^{x}(1+y(t)) \sin t d t $$

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

The autonomous differential equations 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? \(\frac{d P}{d t}=P(1-2 P)\)

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

In Exercises \(7-10,\) write an equivalent first-order differential equation and initial condition for \(y .\) $$ y=1+\int_{0}^{x} y(t) d t $$

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

The autonomous differential equations 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? \(\frac{d P}{d t}=2 P(P-3)\)

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

In Exercises \(11-16,\) 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. $$ y^{\prime}=1-\frac{y}{x}, \quad y(2)=-1, \quad d x=0.5 $$

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

The autonomous differential equations 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? \(\frac{d P}{d t}=3 P(1-P)\left(P-\frac{1}{2}\right)\)

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

In Exercises \(11-16,\) 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. $$ y^{\prime}=x(1-y), \quad y(1)=0, \quad d x=0.2 $$

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.

Salt mixture A tank initially contains 100 gal of brine in which 50 \(\mathrm{lb}\) of salt are dissolved. A brine containing 2 \(\mathrm{lb} / \mathrm{gal}\) of salt runs into the tank at the rate of 5 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 4 gal/min. a. At what rate (pounds per minute) does salt enter the tank at time \(t ?\) b. What is the volume of brine in the tank at time \(t ?\) c. At what rate (pounds per minute) does salt leave the tank at time \(t ?\) d. Write down and solve the initial value problem describing the mixing process. e. Find the concentration of salt in the tank 25 min after the process starts.

In Exercises \(11-16,\) 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. $$ y^{\prime}=2 x y+2 y, \quad y(0)=3, \quad d x=0.2 $$

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/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?

In Exercises \(11-16,\) 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. $$ y^{\prime}=y^{2}(1+2 x), \quad y(-1)=1, \quad d x=0.5 $$

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.) a. Draw a phase line for the equation. b. Sketch a typical velocity curve. c. For a 1110 -lb skydiver \((m g=110)\) and with time in seconds and distance in feet, a typical value of \(k\) is \(0.005 .\) What is the diver's terminal velocity? Repeat for a 200-lb skydiver.

In Exercises \(11-16,\) 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. $$ y^{\prime}=2 x e^{x^{2}}, \quad y(0)=2, \quad d x=0.1 $$

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