Relación con mi mamá o representante (mujer) que me cuida.

In Problems 1–30, solve the equation.

1. 2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23. AyxBdx ^AxyBdy0

A2x 2y8Bdx ^Ax3y6Bdy0

Ay 2x1Bdx ^Axy4Bdy0

dy du y

u 4uy²

Ax²3y²Bdx2xy dy0 dy

dx ^A2xy1B² dy

du 2yy² dy

dxy tan xsin x0 dy

dx2 22xy3 dx

dt x

t1 t²2 dy

dx y

x x²sin 2x

Ay³4e^xyBdx ^A2e^x3y²Bdy0 dx

dt 1cos²AtxB

3^y1

/

² _A1x²2xyy²B¹4^dy0 3^{1 A}1x²2xyy²B¹4^dx

Ax²y²Bdx3xy dy0 dy

dx 2y

x 2x²y² t³y² dtt⁴y⁶ dy0 2xy³ dx A1x²Bdy0

3^{sinAxyBxy cosAxyB}4^dx3^1x2cosAxyB4^dy0 dy

dx 3y

x x²4x3

Ax²2y³Bdy ^A2xy3x²Bdx0

dy

dx 4y32x² dy

dx e^xy y1

1. An instructor at Ivey U. asserted: “All you need to know about first-order differential equations is how to solve those that are exact.” Give arguments that support and arguments that refute the instructor’s claim.

2. What properties do solutions to linear equations have that are not shared by solutions to either sepa-rable or exact equations? Give some specific exam-ples to support your conclusions.

78 Chapter 2 First-Order Differential Equations

TECHNICAL WRITING EXERCISES

3. Consider the differential equation

where a,b, and c are constants. Describe what hap-pens to the asymptotic behavior as of the solution when the constants a, b, and c are varied.

Illustrate with figures and/or graphs.

xSq

dy

dxaybe^x , yA0Bc ,

Oil Spill in a Canal

In 1973 an oil barge collided with a bridge in the Mississippi River, leaking oil into the water at a rate estimated at 50 gallons per minute. In 1989 the Exxon Valdez spilled an estimated 11,000,000 gallons of oil into Prudhoe Bay in 6 hours^†, and in 2010 the Deepwater Horizon well leaked into the Gulf of Mexico at a rate estimated to be 15,000 barrels per day^††(1 barrel = 42 gallons). In this project you are going to use differential equations to analyze a simplified model of the dissipation of heavy crude oil spilled at a rate of Sft³/sec into a flowing body of water. The flow region is a canal, namely a straight channel of rectangular cross section,wfeet wide by dfeet deep, having a constant flow rate of vft/sec; the oil is presumed to float in a thin layer of thickness s(feet) on top of the water, without mixing.

In Figure 2.12, the oil that passes through the cross-section window in a short time t occu-pies a box of dimensions sby wby vt. To make the analysis easier, presume that the canal is conceptually partitioned into cells of length Lft. each, and that within each particular cellthe oil instantaneously disperses and forms a uniformlayer of thickness s_i(t) in cell i(cell 1 starts at the point of the spill). So, at time t, the ith cell contains s_i t wLft³of oil. Oil flows out of cell iat a rate equal to s_i t wvft³/sec, and it flows into cell iat the rate s_i1 t wv; it flows into the first cell at S ft³/sec.

B A B

A A B

79

†Cutler J. Cleveland (Lead Author); C Michael Hogan and Peter Saundry (Topic Editor). 2010. “Deepwater Horizon oil spill.” In:Encyclopedia of Earth, ed. Cutler J. Cleveland (Washington, D.C.: Environmental Information Coalition, National Council for Science and the Environment).

††Cutler J. Cleveland (Contributing Author); National Oceanic and Atmospheric Administration (Content source);

Peter Saundry (Topic Editor). 2010. “Exxon Valdez oil spill.” In:Encyclopedia of Earth, ed. Cutler J. Cleveland (Washington, D.C.: Environmental Information Coalition, National Council for Science and the Environment).

Group Projects for Chapter 2

A

Figure 2.12 Oil leak in a canal.

d

s

w

(a) Formulate a system of differential equations and initial conditions for the oil thickness in the first three cells. Take S50 gallons/min, which was roughly the spillage rate for the Mississippi River incident, and take w200 ft,d 25 ft, and v1 mi/hr (which are reasonable estimates for the Mississippi River^†). Take L1000 ft.

(b) Solve for [Caution: Make sure your units are consistent.]

(c) If the spillage lasts for Tseconds, what is the maximum oil layer thickness in cell 1?

(d) Solve for What is the maximum oil layer thickness in cell 2?

(e) Probably the least tenable simplification in this analysis lies in regarding the layer thickness as uniform over distances of length L. Reevaluate your answer to part (c) with Lreduced to 500 ft. By what fraction does the answer change?

s₂AtB. s₁AtB.

80 Chapter 2 First-Order Differential Equations

Differential Equations in Clinical Medicine

B

Figure 2.13 Lung ventilation pressures lung elastic pressure, Pe , and residual pressure, Pex

airway-resistance pressure drop, Pr

Papp during inspiration Courtesy of Philip Crooke, Vanderbilt University

In medicine,mechanical ventilationis a procedure that assists or replaces spontaneous breath-ing for critically ill patients, usbreath-ing a medical device called a ventilator. Some people attribute the first mechanical ventilation to Andreas Vesalius in 1555. Negative pressure ventilators (iron lungs) came into use in the 1940s–1950s in response to poliomyelitis (polio) epidemics. Philip Drinker and Louis Shaw are credited with its invention. Modern ventilators use positive pressure to inflate the lungs of the patient. In the ICU (intensive care unit), common indications for the ini-tiation of mechanical ventilation are acute respiratory failure, acute exacerbation of chronic obstructive pulmonary disease, coma, and neuromuscular disorders. The goals of mechanical ventilation are to provide oxygen to the lungs and to remove carbon dioxide.

In this project, we model the mechanical process performed by the ventilator. We make the following assumptions about this process of filling the lungs with air and then letting them deflate to some rest volume (see Figure 2.13).

(i) The length (in seconds) of each breath is fixed (t_tot) and is set by the clinician, with each breath being identical to the previous breath.

(ii) Each breath is divided into two parts:inspiration(air flowing into the patient) and expiration(air flowing out of the patient). We assume that inspiration takes place over the interval [0,t_i] and expiration over the time interval [t_i,t_tot]. The time t_iis called the inspiratory time.

†http://www.nps.gov/miss/riverfacts.htm

(iii) During inspiration the ventilator applies a constant pressure P_appto the patient’s air-way, and during expiration this pressure is zero, relative to atmospheric pressure. This is called pressure-controlled ventilation.

(iv) We assume that the pulmonary system (lung) is modeled by a single compartment.

Hence, the action of the ventilator is similar to inflating a balloon and then releasing the pressure.

(v) At the airway there is a pressure balance:

(1)

where denotes pressure losses due to resistance to flow into and out of the lung, is the elastic pressure due to changes in volume of the lung, is a residual pressure that remains in the lung at the completion of a breath, and denotes the pressure applied to the airway. ( during inspiration and during expiration.) The residual pressure is called the end-expiratory pressure.

(vi) Let denote the volume of the lung at time t, with denoting its volume during inspiration and its volume during expiration. We

assume that The number is called the tidal volumeof

the breath.

(vii) We assume that the resistive pressure is proportional to the flows into and out of the lung such that , and we assume that the proportionality constant R is the same for inspiration and expiration.

(viii)Furthermore, we assume that the elastic pressure is proportional to the instantaneous volume of the lung. That is, where the constant C is called the compliance of the lung.

Using the pressure equation in (1) together with the above assumptions, a mathematical model for the instantaneous volume in the single compartment lung is given by the following pair of first-order linear differential equations:

(2)

(3)

The initial conditions, as indicated in assumption (vi), are and . The constant is not known a prioribut is determined from the end condition on the expiratory volume: . This will make each breath identical to the previous breath. To obtain a for-mula for , complete the following steps.

(a) Solve equation (2) for with the initial condition

(b) Solve equation (3) for with the initial condition . (c) Using the fact that , show that

(d) For R10 cm (H₂O)/L/sec,C0.02 L/cm (H₂O),P_app20 cm (H₂O),t_i1 sec and t_tot3 sec, plot the graphs of V_i(t) and V_e(t) over the interval [0,t_tot]. Compute P_exfor

P_ex (e^ti/RC1) P_app e^ttot/RC1 .

V_i(t_i)V_T

V_e(t_i)V_T V_e(t)

V_i(0)0.

V_i(t) P_ex

V_e(t_tot)0 P_ex

V_e(t_i)V_i(t_i)V_T V_i(0)0

Ra^dV_dt^eb a_C¹b^VeP_ex0 , t_itt_tot. Ra^dV_dtⁱb a_C¹b^ViP_exP_app , 0tt_i ,

P_e(1/C)V, P_rR(dV/dt)

P_r

V_iAt_iBV_T V_iA0BV_eAt_totB 0.

V_eAtB, t_i tt_tot,

V_iAtB, 0tt_i, VAtB

P_ex

P_aw0 P_awP_app

P_aw P_ex

P_e P_r

P_rP_eP_exP_aw ,

Group Projects for Chapter 2 81

these parameters.

(e) The mean alveolar pressureis the average pressure in the lung during inspiration and is given by the formula

.

Compute this quantity using your expression for V_i(t) in part (a).

Torricelli’s Law of Fluid Flow

Courtesy of Randall K. Campbell-Wright

How long does it take for water to drain through a hole in the bottom of a tank? Consider the tank pictured in Figure 2.14, which drains through a small, round hole. Torricelli’s law^†states that when the surface of the water is at a height h, the water drains with the velocity it would have if it fell freely from a height h(ignoring various forms of friction).

(a) Show that the standard gravity differential equation

leads to the conclusion that an object that falls from a height will land with a velocity of

(b) Let A(h) be the cross-sectional area of the water in the tank at height hand athe area of the drain hole. The rate at which water is flowing out of the tank at time tcan be expressed as the cross-sectional area at height htimes the rate at which the height of the water is changing. Alternatively, the rate at which water flows out of the hole can be expressed as the area of the hole times the velocity of the draining water. Set these two equal to each other and insert Torricelli’s law to derive the differential equation

(4)

(c) The conical tank of Figure 2.14 has a radius of 30 cm when it is filled to an initial depth of 50 cm. A small round hole at the bottom has a diameter of 1 cm. Determine and aand then solve the differential equation in (4), thus deriving a formula relating time and the height of the water in this tank.

AAhB AAhBdh

dt a22gh . 22ghA0B.

hA0B d²h

dt² g P_m 1

t_i

0^tia^Vi_C^(t)b^dtP_{ex .} 82 Chapter 2 First-Order Differential Equations

C

Figure 2.14 Conical tank A(h)

a 50 cm

30 cm

h

†Historical Footnote:Evangelista Torricelli (1608–1647) invented the barometer and worked on computing the value of the acceleration of gravity as well as observing this principle of fluid flow.

(d) Use your solution to (c) to predict how long it will take for the tank to drain entirely.

(e) Which would drain faster, the tank pictured or an upside-down conical tank of the same dimensions draining through a hole of the same size (1-cm diameter)? How long would it take to drain the upside-down tank?

(f ) Find a water tank and time how long it takes to drain. (You may be able to borrow a

“separatory funnel” from your chemistry department or use a large water cooler.) The tank should be large enough to take several minutes to drain, and the drain hole should be large enough to allow water to flow freely. The top of the tank should be open (so that the water will not “glug”). Repeat steps (c) and (d) for your tank and compare the prediction of Torricelli’s law to your experimental results.

The Snowplow Problem

To apply the techniques discussed in this chapter to real-world problems, it is necessary to trans-late these problems into questions that can be answered mathematically. The process of reformu-lating a real-world problem as a mathematical one often requires making certain simplifying assumptions. To illustrate this, consider the following snowplow problem:

One morning it began to snow very hard and continued snowing steadily throughout the day. A snowplow set out at 9:00 A.M. to clear a road, clearing 2 mi by 11:00 A.M. and an additional mile by 1:00 P.M.At what time did it start snowing?

To solve this problem, you can make two physical assumptions concerning the rate at which it is snowing and the rate at which the snowplow can clear the road. Because it is snowing steadily, it is reasonable to assume it is snowing at a constant rate. From the data given (and from our experience), the deeper the snow, the slower the snowplow moves. With this in mind, assume that the rate (in mph) at which a snowplow can clear a road is inversely proportional to the depth of the snow.

Two Snowplows

Courtesy of Alar Toomre, Massachusetts Institute of Technology

One day it began to snow exactly at noon at a heavy and steady rate. A snowplow left its garage at 1:00 P.M., and another one followed in its tracks at 2:00 P.M. (see Figure 2.15 on page 84).

(a) At what time did the second snowplow crash into the first?To answer this question, assume as in Project D that the rate (in mph) at which a snowplow can clear the road is inversely proportional to the depth of the snow (and hence to the time elapsed since the road was clear of snow). [Hint:Begin by writing differential equations for x(t) and y(t), the distances traveled by the first and second snowplows, respectively, at thours past noon. To solve the differential equation involving y, let trather than ybe the dependent variable!]

(b) Could the crash have been avoided by dispatching the second snowplow at 3:00 P.M. instead?

Group Projects for Chapter 2 83

D

E

Clairaut Equations and Singular Solutions

An equation of the form (5)

where the continuously differentiable function is evaluated at , is called a Clairaut equation.^† Interest in these equations is due to the fact that (5) has a one-parameter family of solutions that consist of straight lines. Further, the envelopeof this family—that is, the curve whose tangent lines are given by the family—is also a solution to (5) and is called the singular solution.

To solve a Clairaut equation:

(a) Differentiate equation (5) with respect to xand simplify to show that (6)

(b) From (6), conclude that or Assume that and

substitute back into (5) to obtain the family of straight-line solutions

(c) Show that another solution to (5) is given parametrically by

where the parameter .This solution is the singular solution.

(d) Use the above method to find the family of straight-line solutions and the singular solu-tion to the equasolu-tion

Here Sketch several of the straight-line solutions along with the singular solution on the same coordinate system. Observe that the straight-line solutions are all tangent to the singular solution.

(e) Repeat part (d) for the equation x(dy/dx)³y(dy/dx)²20 .

fAtB 2t². yxa^dy_dxb 2a^dy_dxb^{2 .}

pdy

/

^dx

yfApBpf¿^ApB , x f¿^ApB , ycxfAcB .

dy

/

^dxc

f¿^Ady

/

^{dxB x.}

dy

/

^dxc

3^xf¿^Ady

/

^dxB4d

2y

dx² 0 , where f¿^AtB d dt fAtB . tdy

/

^dx

fAtB yxdy

dxfAdy

/

^{dxB ,}

84 Chapter 2 First-Order Differential Equations

0 y(t) x(t) Miles from garage

Figure 2.15 Method of successive snowplows

†Historical Footnote:These equations were studied by Alexis Clairaut in 1734.

F

Multiple Solutions of a First-Order Initial Value Problem

Courtesy of Bruce W. Atkinson, Samford University The initial value problem (IVP),

(7)

which was discussed in Example 9 and Problem 29 of Section 1.2, is an example of an IVP that has more than one solution. The goal of this project is to find all the solutions to (7) on ( ). It turns out that there are infinitely many! These solutions can be obtained by con-catenating the three functions for the constant 0 for and for

where as can be seen by completing the following steps:

(a) Show that if is a solution to the differential equation that is not zeroon an open interval I,then on this interval must be of the form

for some constant cnot in I.

(b) Prove that if is a solution to the differential equation on

( , ) and where then for [Hint:

Consider the sign of .]

(c )Now let be a solution to the IVP (7) on ( ). Of course If vanishes at some point then let bbe the largest of such points; otherwise, set

. Similarly, if gvanishes at some point then let abe the smallest (furthest to the left) of such points; otherwise, set . Here we allow and . (Because gis a continuous function, it can be proved that there always exist such largest and smallest points.) Using the results of parts (a) and (b) prove that if both aand bare finite, then ghas the following form:

What is the form of gif ? If ? If both and ?

(d) Verify directly that the above concatenated function g is indeed a solution to the IVP (7) for all choices of aand bwith Also sketch the graph of several of the solu-tion funcsolu-tion gin part (c) for various values of aand b,including infinite values.

We have analyzed here a first-order IVP that not only fails to have a unique solution but has a solution set consisting of a doubly infinite family of functions (with aand bas the two parameters).

Utility Functions and Risk Aversion

Courtesy of James E. Foster, George Washington University

Would you rather have $5 with certainty or a gamble involving a 50% chance of receiving $1 and a 50% chance of receiving $11? The gamble has a higher expected value ($6); however, it also has a greater level of risk. Economists model the behavior of consumers or other agents facing

a2b.

a q

b q

a q

b q

g(x)

^{(x0(xa)b)33 if xif aif x76abxb .}

a q

b q

a2 x 6 2, b2

x 7 2,

g g(2)0.

q, q yg(x)

f¿

axb.

f(x)0 a 6 b,

f(a)f(b)0, q

q

dy/dx3y^2/3 yf(x)

f(x)(xc)³

f(x)

dy/dx3y^2/3 yf(x)

a2b, x 7 b,

(xb)³ axb,

x 6 a, (xa)³

q, q dy

dx3y^2/3 , yA2B 0 ,

Group Projects for Chapter 2 85

G

H

risky decisions with the help of a (von Neumann–Morgenstern) utility function uand the criterion of expected utility.

Rather than using expected values of the dollar payoffs, the payoffs are first transformed into utility levels and then weighted by probabilities to obtain expected utility. Following the sugges-tion of Daniel Bernoulli, we might set and then compare to [0.5 ln 1 0.5 ln 11] 0.1969, which would result in the sure thing being chosen in this case rather than the gamble. This utility function is strictly concave, which corresponds to the agent being risk averse, or wanting to avoid gambles (unless of course the extra risk is sufficiently compensated by a high enough increase in the mean or expected payoff ).

Alternatively, the utility function might be , which is strictly convex and corre-sponds to the agent being risk loving.This agent would surely select the above gamble. The case of occurs when the agent is risk neutral and would select according to the expected value of the payoff. It is normally assumed that at all payoff levels,x;in other words, higher payoffs are desirable.

In addition to knowing if an agent is risk averse or risk loving, economists are often inter-ested in knowing howrisk averse (or risk loving) an agent is. Clearly this has something to do with the second derivative of the utility function. The measure of relative risk aversion of an agent with utility function and payoff xis defined as . Normally, is a function of the payoff level. However, economists often find it convenient to restrict considera-tion to utility funcconsidera-tions for which is constant, say, for all x.It is easily shown that each of u(x) ln x, u(x) x², and exhibits constant relative risk aversion (with levels and respectively). A question naturally arises: What is the set of all utility functions that have constant relative risk aversion?

(a) State the second-order differential equation defined by the above question.

(b) Convert this into a separable first-order differential equation for solve, and use the solution to determine the possible forms that can take.

(c) Integrate to obtain the set of all constant relative risk-aversion utility functions. This class is used extensively throughout economics.

(d) An alternative measure of risk aversion is the measure of absolute risk aversion. Find the set of all utility functions exhibiting constant absolute risk aversion.

(e) Which functions are both constant absoluteand constant relativerisk-aversion utility functions?

For further reading, see, for example, the economics text Microeconomic Theory, by A. Mas-Colell, M. Whinston, and J. Green (Oxford University Press, Oxford, 1995).

Designing a Solar Collector

You want to design a solar collector that will concentrate the sun’s rays at a point. By symmetry this surface will have a shape that is a surface of revolution obtained by revolving a curve about an axis. Without loss of generality, you can assume that this axis is the x-axis and the rays parallel to this axis are focused at the origin (see Figure 2.16). To derive the equation for the curve, proceed as follows:

(a) The law of reflection says that the angles and are equal. Use this and results from geometry to show that b2a.

d g u(x)

a(x)u–(x)

/

^u¿(x),

u¿(x)

u¿(x), s0,

s1, s 1,

u(x)x

r(x)s r(x)

r(x) r(x) u–(x)x

/

^u¿(x)

u(x)

u¿(x) 7 0 u(x)x

u(x)x²

ln 50.8047 u(x)ln x

86 Chapter 2 First-Order Differential Equations

I

(b) From calculus recall that . Use this, the fact that , and the double angle formula to show that

(c) Now show that the curve satisfies the differential equation (8)

(d) Solve equation (8). [Hint:See Section 2.6.]

(e) Describe the solutions and identify the type of collector obtained.

dy

dxx 2x²y²

y .

y

x 2 dy/dx 1(dy/dx)² .

y/xtan b dy/dxtan a

Group Projects for Chapter 2 87

Figure 2.16 Curve that generates a solar collector x tangent line

unknown curve

sun rays y

0

α β

γ (x, y) δ

Asymptotic Behavior of Solutions to Linear Equations

To illustrate how the asymptotic behavior of the forcing term affects the solution to a linear equation, consider the equation

(9)

where the constant ais positive and is continuous on .

(a) Show that the general solution to equation (9) can be written in the form

where x₀is a nonnegative constant.

(b) If for , where kand are nonnegative constants, show that 0yAxB0 0yAx₀B0e^aAxx0B k

a3¹e^aAxx0B4 for xx₀ . x₀

xx₀

0QAxB0 k

yAxB yAx₀Be^aAxx0Be^ax

x^x0

e^atQAtBdt ,

3^0, q^B QAxB

dy

dxayQAxB ,

QAxB

J

(c) Let satisfy the same equation as (9) but with forcing function . That is,

where is continuous on Show that if

then

(d) Now show that if as , then any solution of equation (9) satisfies

[Hint:Take in part (c).]

(e) As an application of part (d), suppose a brine solution containing kg of salt per liter at time truns into a tank of water at a fixed rate and that the mixture, kept uniform by stirring, flows out at the same rate. Given that use the result of part (d) to determine the limiting concentration of the salt in the tank as (see Exer-cises 2.3, Problem 35).

tS q qAtBSb as tS q,

qAtB Q~_A

xBb and zAxBb

/

^a

yAxBSb

/

^{a as xS q.}

yAxB xS q

QAxBSb

0zAxB yAxB0 0zAx₀ByAx₀B0e^aAxx0B K

a 3¹e^aAxx0B4 for x x₀ . 0Q~_A

xBQAxB0 K for x x₀ , 3^0, q^B. Q~_A

xB dz

dxazQ~_A xB ,

Q~_A xB zAxB

88 Chapter 2 First-Order Differential Equations

89 Adopting the Babylonian practices of careful measurement and detailed observations, the ancient Greeks sought to comprehend nature by logical analysis. Aristotle’s convincing arguments that the world was not flat, but spherical, led the intellectuals of that day to ponder the question: What is the circumference of Earth? And it was astonishing that Eratosthenes managed to obtain a fairly accurate answer to this problem without having to set foot beyond the ancient city of Alexandria. His method involved certain assumptions and simplifications:

Earth is a perfect sphere, the Sun’s rays travel parallel paths, the city of Syene was 5000 stadia due south of Alexandria, and so on. With these idealizations Eratosthenes created a mathemati-cal context in which the principles of geometry could be applied.^†

Today, as scientists seek to further our understanding of nature and as engineers seek, on a more pragmatic level, to find answers to technical problems, the technique of repre-senting our “real world” in mathematical terms has become an invaluable tool. This process of mimicking reality by using the language of mathematics is known as mathematical modeling.

Formulating problems in mathematical terms has several benefits. First, it requires that we clearly state our premises. Real-world problems are often complex, involving several different and possibly interrelated processes. Before mathematical treatment can proceed, one must determine which variables are significant and which can be ignored. Often, for the relevant variables, relationships are postulated in the form of laws, formulas, theories, and the like.

These assumptions constitute the idealizationsof the model.

Mathematics contains a wealth of theorems and techniques for making logical deductions and manipulating equations. Hence, it provides a context in which analysis can take place free of any preconceived notions of the outcome. It is also of great practical importance that mathe-matics provides a format for obtaining numerical answers via a computer.

The process of building an effective mathematical model takes skill, imagination, and objective evaluation. Certainly an exposure to several existing models that illustrate various aspects of modeling can lead to a better feel for the process. Several excellent books and articles are devoted exclusively to the subject.^††In this chapter we concentrate on examples of

Mathematical Models and

Numerical Methods Involving First-Order Equations

CHAPTER 3

In document Estudio sobre las familias migrantes y la incidencia en las relaciones escolares y familiares de los hijos, realizado en el sexto y séptimo año de educación general básica, de la escuela dos de agosto de la parroquia de ciano cantón Puyango provincia de L (página 76-85)