1.1 Exercises Answers to selected odd-numbered problems begin on page ANS-1.

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1 . Eercises Answers to selected odd-numbered problems begin on page ANS-. In Problems 8, state the order of the given ordinar differential equation. Determine whether the equation is linear or nonlinear b matching it with (6).. ( ) 4 cos d d 2 ad 3 d b 0 3. t (4) t d 2 u dr du u cos (r u) 2 dr. d 2 2 d 2 Å ad d b 6. d 2 R dt 2 k R 2 7. (sin u) (cos u) 2 8. $ 2 ( 2 3 # 2 ) # 0 In Problems 9 and 0, determine whether the given first-order differential equation is linear in the indicated dependent variable b matching it with the first differential equation given in (7). 9. ( 2 ) d d 0; in ; in 0. u dv (v uv ue u ) du 0; in v; in u In Problems 4, verif that the indicated function is an eplicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.. 2 0; e /2 2. d 20 24; dt e 20t ; e 3 cos 2 4. tan ; (cos ) ln(sec tan ) In Problems 8, verif that the indicated function f() is an eplicit solution of the given first-order differential equation. Proceed as in Eample, b considering f simpl as a function, give its domain. Then b considering f as a solution of the differential equation, give at least one interval I of definition.. ( 2 )9 2 8; 4" ; tan ; /(4 2 ) cos ; ( sin ) /2 In Problems 9 and 20, verif that the indicated epression is an implicit solution of the given first-order differential equation. Find at least one eplicit solution f() in each case. Use a graphing utilit to obtain the graph of an eplicit solution. Give an interval I of definition of each solution f. 9. dx 2 (X 2 )( 2 2X); ln a2x dt X 2 b t d ( 2 ) d 0; In Problems 2 24, verif that the indicated famil of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. 2. dp dt P( 2 P); P c e t c e t 22. d 2 ; d e 23. d 2 d 2 4 d 4 0; 2 d 2# 0 e t2 dt c e 2 c e 2 c 2 e d 3 d 3 22 d 2 d 2 2 d d 2 2 ; c c 2 c 3 ln Verif that the piecewise-defined function e 2,, 0 2, $ 0 is a solution of the differential equation 2 0 on the interval ( q, q). 26. In Eample 6 we saw that f () "2 2 2 and f 2 () "2 2 2 are solutions of d/d / on the interval (, ). Eplain wh the piecewise-defined function e "2 2 2,,, 0 "2 2 2, 0 #, is not a solution of the differential equation on the interval (, ). In Problems 27 30, find values of m so that the function e m is a solution of the given differential equation In Problems 3 and 32, find values of m so that the function m is a solution of the given differential equation In Problems 33 36, use the concept that c, q q, is a constant function if and onl if 0 to determine whether the given differential equation possesses constant solutions ( ) CHAPTER Introduction to Differential Equations

2 In Problems 37 and 38, verif that the indicated pair of functions is a solution of the given sstem of differential equations on the interval ( q, q). 37. d dt 3, 38. d 2 dt 2 4 et, d d 2 3; dt dt et ; e 2t 3e 6t, cos 2t sin 2t et, e 2t e 6t Discussion Problems cos 2t sin 2t et 39. Make up a differential equation that does not possess an real solutions. 40. Make up a differential equation that ou feel confident possesses onl the trivial solution 0. Eplain our reasoning. 4. What function do ou know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple k of itself? Write each answer in the form of a first-order differential equation with a solution. 42. What function (or functions) do ou know from calculus is such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a second-order differential equation with a solution. 43. Given that sin is an eplicit solution of the first-order differential equation d/d " 2 2. Find an interval I of definition. [Hint: I is not the interval ( q, q).] 44. Discuss wh it makes intuitive sense to presume that the linear differential equation 2 4 sin t has a solution of the form Asin t Bcos t, where A and B are constants. Then find specific constants A and B so that Asin t Bcos t is a particular solution of the DE. In Problems 4 and 46, the given figure represents the graph of an implicit solution G(, ) 0 of a differential equation d/d f (, ). In each case the relation G(, ) 0 implicitl defines several solutions of the DE. Carefull reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution f must be a function and differentiable. Use the solution curve to estimate the interval I of definition of each solution f. 4. FIGURE.. Graph for Problem FIGURE..6 Graph for Problem The graphs of the members of the one-parameter famil 3 3 3c are called folia of Descartes. Verif that this famil is an implicit solution of the first-order differential equation d d ( ) ( ). 48. The graph in FIGURE..6 is the member of the famil of folia in Problem 47 corresponding to c. Discuss: How can the DE in Problem 47 help in finding points on the graph of where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining an interval I of definition of a solution f of the DE? Carr out our ideas and compare with our estimates of the intervals in Problem In Eample 6, the largest interval I over which the eplicit solutions f () and f 2 () are defined is the open interval (, ). Wh can t the interval I of definition be the closed interval [, ]? 0. In Problem 2, a one-parameter famil of solutions of the DE P P( P) is given. Does an solution curve pass through the point (0, 3)? Through the point (0, )?. Discuss, and illustrate with eamples, how to solve differential equations of the forms d/d f () and d 2 /d 2 f (). 2. The differential equation ( ) has the form given in (4). Determine whether the equation can be put into the normal form d/d f (, ). 3. The normal form () of an nth-order differential equation is equivalent to (4) whenever both forms have eactl the same solutions. Make up a first-order differential equation for which F(,, ) 0 is not equivalent to the normal form d/d f (, ). 4. Find a linear second-order differential equation F(,,, ) 0 for which c c 2 2 is a two-parameter famil of solutions. Make sure that our equation is free of the arbitrar parameters c and c 2. Qualitative information about a solution f() of a differential equation can often be obtained from the equation itself. Before working Problems 8, recall the geometric significance of the derivatives d/d and d 2 /d 2.. Consider the differential equation d/d e 2. (a) Eplain wh a solution of the DE must be an increasing function on an interval of the -ais. (b) What are lim d/d and limd/d? What does this S 2 q Sq suggest about a solution curve as S q? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution f() of the differential equation whose shape is suggested b parts (a) (c). 6. Consider the differential equation d/d. (a) Either b inspection, or b the method suggested in Problems 33 36, find a constant solution of the DE. (b) Using onl the differential equation, find intervals on the -ais on which a nonconstant solution f() is increasing. Find intervals on the -ais on which f() is decreasing.. Definitions and Terminolog

3 7. Consider the differential equation d/d (a b), where a and b are positive constants. (a) Either b inspection, or b the method suggested in Problems 33 36, find two constant solutions of the DE. (b) Using onl the differential equation, find intervals on the -ais on which a nonconstant solution f() is increasing. On which f() is decreasing. (c) Using onl the differential equation, eplain wh a/2b is the -coordinate of a point of inflection of the graph of a nonconstant solution f(). (d) On the same coordinate aes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the -plane into three regions. In each region, sketch the graph of a nonconstant solution f() whose shape is suggested b the results in parts (b) and (c). 8. Consider the differential equation 2 4. (a) Eplain wh there eist no constant solutions of the DE. (b) Describe the graph of a solution f(). For eample, can a solution curve have an relative etrema? (c) Eplain wh 0 is the -coordinate of a point of inflection of a solution curve. (d) Sketch the graph of a solution f() of the differential equation whose shape is suggested b parts (a) (c). Computer Lab Assignments In Problems 9 and 60, use a CAS to compute all derivatives and to carr out the simplifications needed to verif that the indicated function is a particular solution of the given differential equation. 9. (4) ; e cos ; 20 cos ( ln ) 2 3 sin ( ln ) 2 CHAPTER Introduction to Differential Equations

4 .2 Eercises Answers to selected odd-numbered problems begin on page ANS-. In Problems and 2, /( c e ) is a one-parameter famil of solutions of the first-order DE 2. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.. (0) 3 2. ( ) 2 In Problems 3 6, /( 2 c) is a one-parameter famil of solutions of the first-order DE Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. 3. (2) 3 4. ( 2) 2. (0) 6. ( 2) 4 In Problems 7 0, c cos t c 2 sin t is a two-parameter famil of solutions of the second-order DE 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 7. (0), (0) 8 8. (p/2) 0, (p/2) 9. (p/6) 2, (p/6) 0 0. (p/4) "2, (p/4) 2"2 In Problems 4, c e c 2 e is a two-parameter famil of solutions of the second-order DE 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.. (0), (0) 2 2. () 0, () e 3. ( ), ( ) 4. (0) 0, (0) 0 In Problems and 6, determine b inspection at least two solutions of the given first-order IVP.. 3 2/3, (0) , (0) 0 In Problems 7 24, determine a region of the -plane for which the given differential equation would have a unique solution whose graph passes through a point ( 0, 0 ) in the region. 7. d d d 2/3 8. d " 9. d d 20. d d 2 2. (4 2 2 ) ( 3 ) ( 2 2 ) ( 2 )9 In Problems 2 28, determine whether Theorem.2. guarantees that the differential equation 9 " possesses a unique solution through the given point. 2. (, 4) 26. (, 3) 27. (2, 3) 28. (, ) 29. (a) B inspection, find a one-parameter famil of solutions of the differential equation. Verif that each member of the famil is a solution of the initial-value problem, (0) 0. (b) Eplain part (a) b determining a region R in the -plane for which the differential equation would have a unique solution through a point ( 0, 0 ) in R. (c) Verif that the piecewise-defined function e 0,, 0, $ 0 satisfies the condition (0) 0. Determine whether this function is also a solution of the initial-value problem in part (a). 30. (a) Verif that tan ( c) is a one-parameter famil of solutions of the differential equation 2. (b) Since f (, ) 2 and f/ 2 are continuous everwhere, the region R in Theorem.2. can be taken to be the entire -plane. Use the famil of solutions in part (a) to find an eplicit solution of the first-order initialvalue problem 2, (0) 0. Even though 0 0 is in the interval ( 2, 2), eplain wh the solution is not defined on this interval. (c) Determine the largest interval I of definition for the solution of the initial-value problem in part (b). 3. (a) Verif that /( c) is a one-parameter famil of solutions of the differential equation 2. (b) Since f (, ) 2 and f/ 2 are continuous everwhere, the region R in Theorem.2. can be taken to be the entire -plane. Find a solution from the famil in part (a) that satisfies (0). Find a solution from the famil in part (a) that satisfies (0). Determine the largest interval I of definition for the solution of each initial-value problem. 32. (a) Find a solution from the famil in part (a) of Problem 3 that satisfies 2, (0) 0, where 0 0. Eplain wh the largest interval I of definition for this solution is either ( q, / 0 ) or (/ 0, q). (b) Determine the largest interval I of definition for the solution of the first-order initial-value problem 2, (0) (a) Verif that c is a one-parameter famil of solutions of the differential equation d/d 3. (b) B hand, sketch the graph of the implicit solution Find all eplicit solutions f() of the DE in part (a) defined b this relation. Give the interval I of definition of each eplicit solution. (c) The point ( 2, 3) is on the graph of , but which of the eplicit solutions in part (b) satisfies ( 2) 3? 6 CHAPTER Introduction to Differential Equations

5 34. (a) Use the famil of solutions in part (a) of Problem 33 to find an implicit solution of the initial-value problem d/d 3, (2) 4. Then, b hand, sketch the graph of the eplicit solution of this problem and give its interval I of definition. (b) Are there an eplicit solutions of d/d 3 that pass through the origin? In Problems 3 38, the graph of a member of a famil of solutions of a second-order differential equation d 2 /d 2 f (,, ) is given. Match the solution curve with at least one pair of the following initial conditions. (a) (), () 2 (b) ( ) 0, ( ) 4 (c) (), () 2 (d) (0), (0) 2 (e) (0), (0) 0 (f) (0) 4, (0) 2 Discussion Problems In Problems 4 and 46, use Problem in Eercises. and (2) and (3) of this section. 4. Find a function f () whose graph at each point (, ) has the slope given b 8e 2 6 and has the -intercept (0, 9). 46. Find a function f () whose second derivative is 2 2 at each point (, ) on its graph and is tangent to the graph at the point corresponding to. 47. Consider the initial-value problem 2, (0) 2. Determine which of the two curves shown in FIGURE.2. is the onl plausible solution curve. Eplain our reasoning. (0, ¹ ₂) FIGURE.2. Graph for Problem FIGURE.2.7 Graph for Problem FIGURE.2.8 Graph for Problem Determine a plausible value of 0 for which the graph of the solution of the initial-value problem 2 3 6, ( 0 ) 0 is tangent to the -ais at ( 0, 0). Eplain our reasoning. 49. Suppose that the first-order differential equation d/d f (, ) possesses a one-parameter famil of solutions and that f (, ) satisfies the hpotheses of Theorem.2. in some rectangular region R of the -plane. Eplain wh two different solution curves cannot intersect or be tangent to each other at a point ( 0, 0 ) in R. 0. The functions () 6 4, q,, q and () e 0,, 0 6 4, $ 0 FIGURE.2.9 Graph for Problem 37 FIGURE.2.0 Graph for Problem 38 In Problems 39 44, c cos 3 c 2 sin 3 is a two-parameter famil of solutions of the second-order DE 9 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundar conditions. 39. (0) 0, (p/6) 40. (0) 0, (p) 0 4. (0) 0, (p/4) (0), (p) 43. (0) 0, (p) (p/3), (p) 0 have the same domain but are clearl different. See FIGURES.2.2(a) and.2.2(b), respectivel. Show that both functions are solutions of the initial-value problem d/d /2, (2) on the interval ( q, q). Resolve the apparent contradiction between this fact and the last sentence in Eample. (a) (2, ) FIGURE.2.2 Two solutions of the IVP in Problem 0 (b) (2, ).2 Initial-Value Problems 7

6 Mathematical Model. Population Growth Beginning in the net section we will see that differential equations can be used to describe or model man different phsical sstems. In this problem, suppose that a model of the growing population of a small communit is given b the initial-value problem where P is the number of individuals in the communit and time t is measured in ears. How fast, that is, at what rate, is the population increasing at t 0? How fast is the population increasing when the population is 00? dp dt 0.P(t) 20, P(0) 00, 8 CHAPTER Introduction to Differential Equations

7 .3 Eercises Answers to selected odd-numbered problems begin on page ANS-. Population Dnamics. Under the same assumptions underling the model in (), determine a differential equation governing the growing population P(t) of a countr when individuals are allowed to immigrate into the countr at a constant rate r 0. What is the differential equation for the population P(t) of the countr when individuals are allowed to emigrate at a constant rate r 0? 2. The population model given in () fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a communit, it is assumed that the rate at which the population changes is a net rate that is, the difference between the rate of births and the rate of deaths in the communit. Determine a model for the population P(t) if both the birth rate and the death rate are proportional to the population present at time t. 3. Using the concept of a net rate introduced in Problem 2, determine a differential equation governing a population P(t) if the birth rate is proportional to the population present at time t but the death rate is proportional to the square of the population present at time t. 4. Modif the model in Problem 3 for the net rate at which the population P(t) of a certain kind of fish changes, b also assuming that the fish are harvested at a constant rate h 0. Newton s Law of Cooling/Warming. A cup of coffee cools according to Newton s law of cooling (3). Use data from the graph of the temperature T(t) in FIGURE.3.0 to estimate the constants T m, T 0, and k in a model of the form of the first-order initial-value problem T dt dt k(t 2 T m), T(0) T t min FIGURE.3.0 Cooling curve in Problem 6. The ambient temperature T m in (3) could be a function of time t. Suppose that in an artificiall controlled environment, T m (t) is periodic with a 24-hour period, as illustrated in FIGURE.3.. Devise a mathematical model for the temperature T(t) of a bod within this environment. T m (t) t Midnight Noon Midnight Noon Midnight FIGURE.3. Ambient temperature in Problem 6 Spread of a Disease/Technolog 7. Suppose a student carring a flu virus returns to an isolated college campus of 000 students. Determine a differential equation governing the number of students (t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number of students with the flu and the number of students who have not et been eposed to it. 8. At a time t 0, a technological innovation is introduced into a communit with a fied population of n people. Determine a differential equation governing the number of people (t) who have adopted the innovation at time t if it is assumed that the rate at which the innovation spreads through the communit is jointl proportional to the number of people who have adopted it and the number of people who have not adopted it. Mitures 9. Suppose that a large miing tank initiall holds 300 gallons of water in which 0 pounds of salt has been dissolved. Pure water is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is pumped out at the same rate. Determine a differential equation for the amount A(t) of salt in the tank at time t. What is A(0)? 0. Suppose that a large miing tank initiall holds 300 gallons of water in which 0 pounds of salt has been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation for the amount A(t) of salt in the tank at time t.. What is the differential equation in Problem 0, if the well-stirred solution is pumped out at a faster rate of 3. gal/min? 2. Generalize the model given in (8) on page 2 b assuming that the large tank initiall contains N 0 number of gallons of brine, r in and r out are the input and output rates of the brine,.3 Differential Equations as Mathematical Models 2

8 respectivel (measured in gallons per minute), c in is the concentration of the salt in the inflow, c(t) is the concentration of the salt in the tank as well as in the outflow at time t (measured in pounds of salt per gallon), and A(t) is the amount of salt in the tank at time t. Draining a Tank 3. Suppose water is leaking from a tank through a circular hole of area A h at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of the water leaving the tank per second to ca h "2gh, where c (0 c ) is an empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank in FIGURE.3.2. The radius of the hole is 2 in and g 32 ft/s 2. h circular hole A w 0 ft FIGURE.3.2 Cubical tank in Problem 3 6. A series circuit contains a resistor and a capacitor as shown in FIGURE.3.. Determine a differential equation for the charge q(t) on the capacitor if the resistance is R, the capacitance is C, and the impressed voltage is E(t). E R C FIGURE.3. RC-series circuit in Problem 6 Falling Bodies and Air Resistance 7. For high-speed motion through the air such as the skdiver shown in FIGURE.3.6 falling before the parachute is opened air resistance is closer to a power of the instantaneous velocit v(t). Determine a differential equation for the velocit v(t) of a falling bod of mass m if air resistance is proportional to the square of the instantaneous velocit. kv 2 4. The right-circular conical tank shown in FIGURE.3.3 loses water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t. The radius of the hole is 2 in, g 32 ft/s 2, and the friction/contraction factor introduced in Problem 3 is c ft A w 20 ft h circular hole FIGURE.3.3 Conical tank in Problem 4 Series Circuits. A series circuit contains a resistor and an inductor as shown in FIGURE.3.4. Determine a differential equation for the current i(t) if the resistance is R, the inductance is L, and the impressed voltage is E(t). mg FIGURE.3.6 Air resistance proportional to square of velocit in Problem 7 Newton s Second Law and Archimedes Principle 8. A clindrical barrel s ft in diameter of weight w lb is floating in water as shown in FIGURE.3.7(a). After an initial depression, the barrel ehibits an up-and-down bobbing motion along a vertical line. Using Figure.3.7(b), determine a differential equation for the vertical displacement (t) if the origin is taken to be on the vertical ais at the surface of the water when the barrel is at rest. Use Archimedes principle: Buoanc, or upward force of the water on the barrel, is equal to the weight of the water displaced. Assume that the downward direction is positive, that the weight densit of water is 62.4 lb/ft 3, and that there is no resistance between the barrel and the water. s/2 s/2 0 surface 0 (t) E L R (a) (b) FIGURE.3.4 LR-series circuit in Problem FIGURE.3.7 Bobbing motion of floating barrel in Problem 8 26 CHAPTER Introduction to Differential Equations

9 Newton s Second Law and Hooke s Law 9. After a mass m is attached to a spring, it stretches s units and then hangs at rest in the equilibrium position as shown in FIGURE.3.8(b). After the spring/mass sstem has been set in motion, let (t) denote the directed distance of the mass beond the equilibrium position. As indicated in Figure.3.8(c), assume that the downward direction is positive, that the motion takes place in a vertical straight line through the center of gravit of the mass, and that the onl forces acting on the sstem are the weight of the mass and the restoring force of the stretched spring. Use Hooke s law: The restoring force of a spring is proportional to its total elongation. Determine a differential equation for the displacement (t) at time t. unstretched spring (a) s (t) < 0 m = 0 (t) > 0 equilibrium position m (b) FIGURE.3.8 Spring/mass sstem in Problem In Problem 9, what is a differential equation for the displacement (t) if the motion takes place in a medium that imparts a damping force on the spring/mass sstem that is proportional to the instantaneous velocit of the mass and acts in a direction opposite to that of motion? Newton s Second Law and Variable Mass When the mass m of a bod moving through a force field is variable, Newton s second law takes on the form: If the net force acting on a bod is not zero, then the net force F is equal to the time rate of change of momentum of the bod. That is, (c) F d dt (mv)*, (7) where mv is momentum. Use this formulation of Newton s second law in Problems 2 and Consider a single-stage rocket that is launched verticall upward as shown in the accompaning photo. Let m(t) denote the total mass of the rocket at time t (which is the sum of three masses: the constant mass of the paload, the constant mass of the vehicle, and the variable amount of fuel). Assume that the positive direction is upward, air resistance is proportional to the instantaneous velocit v of the rocket, and R is the upward thrust or force generated b the propulsion sstem. *Note that when m is constant, this is the same as F ma. Use (7) to find a mathematical model for the velocit v(t) of the rocket. Rocket in Problem In Problem 2, suppose m(t) m p m v m f (t) where m p is constant mass of the paload, m v is the constant mass of the vehicle, and m f (t) is the variable amount of fuel. (a) Show that the rate at which the total mass of the rocket changes is the same as the rate at which the mass of the fuel changes. (b) If the rocket consumes its fuel at a constant rate l, find m(t). Then rewrite the differential equation in Problem 2 in terms of l and the initial total mass m(0) m 0. (c) Under the assumption in part (b), show that the burnout time t b 0 of the rocket, or the time at which all the fuel is consumed, is t b m f (0)/l, where m f (0) is the initial mass of the fuel. Newton s Second Law and the Law of Universal Gravitation 23. B Newton s law of universal gravitation, the free-fall acceleration a of a bod, such as the satellite shown in FIGURE.3.9, falling a great distance to the surface is not the constant g. Rather, the acceleration a is inversel proportional to the square of the distance from the center of the Earth, a k/r 2, where k is the constant of proportionalit. Use the fact that at the surface of the Earth r R and a g to determine k. If the positive direction is upward, use Newton s second law and his universal law of gravitation to find a differential equation for the distance r. satellite of mass m surface R Earth of mass M r FIGURE.3.9 Satellite in Problem 23.3 Differential Equations as Mathematical Models 27

10 24. Suppose a hole is drilled through the center of the Earth and a bowling ball of mass m is dropped into the hole, as shown in FIGURE Construct a mathematical model that describes the motion of the ball. At time t let r denote the distance from the center of the Earth to the mass m, M denote the mass of the Earth, M r denote the mass of that portion of the Earth within a sphere of radius r, and d denote the constant densit of the Earth. surface a surface of revolution with the propert that all light ras L parallel to the -ais striking the surface are reflected to a single point O (the origin). Use the fact that the angle of incidence is equal to the angle of reflection to determine a differential equation that describes the shape of the curve C. Such a curve C is important in applications ranging from construction of telescopes to satellite antennas, automobile headlights, and solar collectors. [Hint: Inspection of the figure shows that we can write f 2u. Wh? Now use an appropriate trigonometric identit.] R r m tangent C P(, ) θ L FIGURE.3.20 Hole through Earth in Problem 24 Miscellaneous Mathematical Models 2. Learning Theor In the theor of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t. Determine a differential equation for the amount A(t). 26. Forgetfulness In Problem 2, assume that the rate at which material is forgotten is proportional to the amount memorized in time t. Determine a differential equation for A(t) when forgetfulness is taken into account. 27. Infusion of a Drug A drug is infused into a patient s bloodstream at a constant rate of r grams per second. Simultaneousl, the drug is removed at a rate proportional to the amount (t) of the drug present at time t. Determine a differential equation governing the amount (t). 28. Tractri A motorboat starts at the origin and moves in the direction of the positive -ais, pulling a waterskier along a curve C called a tractri. See FIGURE.3.2. The waterskier, initiall located on the -ais at the point (0, s), is pulled b keeping a rope of constant length s, which is kept taut throughout the motion. At time t. 0 the waterskier is at the point (, ) Find the differential equation of the path of motion C. (0, s) waterskier (, ) s motorboat FIGURE.3.2 Tractri curve in Problem 28 θ 29. Reflecting Surface Assume that when the plane curve C shown in FIGURE.3.22 is revolved about the -ais it generates C θ O φ FIGURE.3.22 Reflecting surface in Problem 29 Discussion Problems 30. Reread Problem 4 in Eercises. and then give an eplicit solution P(t) for equation (). Find a one-parameter famil of solutions of (). 3. Reread the sentence following equation (3) and assume that T m is a positive constant. Discuss wh we would epect k 0 in (3) in both cases of cooling and warming. You might start b interpreting, sa, T(t) T m in a graphical manner. 32. Reread the discussion leading up to equation (8). If we assume that initiall the tank holds, sa, 0 lbs of salt, it stands to reason that since salt is being added to the tank continuousl for t 0, that A(t) should be an increasing function. Discuss how ou might determine from the DE, without actuall solving it, the number of pounds of salt in the tank after a long period of time. 33. Population Model The differential equation dp/dt (k cos t)p where k is a positive constant, is a model of human population P(t) of a certain communit. Discuss an interpretation for the solution of this equation; in other words, what kind of population do ou think the differential equation describes? 34. Rotating Fluid As shown in FIGURE.3.23(a), a right-circular clinder partiall filled with fluid is rotated with a constant angular velocit v about a vertical -ais through its center. The rotating fluid is a surface of revolution S. To identif S we first establish a coordinate sstem consisting of a vertical plane determined b the -ais and an -ais drawn perpendicular to the -ais such that the point of intersection of the aes (the origin) is located at the lowest point on the surface S. We then seek a function f (), which represents the curve C of intersection of the surface S and the vertical coordinate plane. Let the point P(, ) denote the position of a 28 CHAPTER Introduction to Differential Equations

11 particle of the rotating fluid of mass m in the coordinate plane. See Figure.3.23(b). (a) At P, there is a reaction force of magnitude F due to the other particles of the fluid, which is normal to the surface S. B Newton s second law the magnitude of the net force acting on the particle is mv 2. What is this force? Use Figure.3.23(b) to discuss the nature and origin of the equations F cos u mg, F sin u mv 2. (b) Use part (a) to find a first-order differential equation that defines the function f (). (a) w P curve C of intersection of -plane and surface of revolution FIGURE.3.23 Rotating fluid in Problem 34 (b) tangent line to curve C at P θ mw 2 P(, ) θ mg 3. Falling Bod In Problem 23, suppose r R s, where s is the distance from the surface of the Earth to the falling bod. What does the differential equation obtained in Problem 23 become when s is ver small compared to R? 36. Raindrops Keep Falling In meteorolog, the term virga refers to falling raindrops or ice particles that evaporate before the reach the ground. Assume that a tpical raindrop is spherical in shape. Starting at some time, which we can designate as t 0, the raindrop of radius r 0 falls from rest from a cloud and begins to evaporate. (a) If it is assumed that a raindrop evaporates in such a manner that its shape remains spherical, then it also makes sense to assume that the rate at which the raindrop evaporates that is, the rate at which it loses mass is proportional to its surface area. Show that this latter assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Find r(t). [Hint: See Problem in Eercises..] (b) If the positive direction is downward, construct a mathematical model for the velocit v of the falling raindrop at time t. Ignore air resistance. [Hint: See the introduction to Problems 2 and 22.] 37. Let It Snow The snowplow problem is a classic and appears in man differential equations tets but was probabl made famous b Ralph Palmer Agnew: One da it started snowing at a heav and stead rate. A snowplow started out at noon, going 2 miles the first hour and mile the second hour. What time did it start snowing? If possible, find the tet Differential Equations, Ralph Palmer Agnew, McGraw-Hill, and then discuss the construction and solution of the mathematical model. 38. Reread this section and classif each mathematical model as linear or nonlinear. 39. Population Dnamics Suppose that P (t) 0. P(t) represents a mathematical model for the growth of a certain cell culture, where P(t) is the size of the culture (measured in millions of cells) at time t (measured in hours). How fast is the culture growing at the time t when the size of the culture reaches 2 million cells? 40. Radioactive Deca Suppose that A (t) A(t) represents a mathematical model for the deca of radium- 226, where A(t) is the amount of radium (measured in grams) remaining at time t (measured in ears). How much of the radium sample remains at time t when the sample is decaing at a rate of grams per ear? Chapter in Review Answers to selected odd-numbered problems begin on page ANS-2. In Problems and 2, fill in the blank and then write this result as a linear first-order differential equation that is free of the smbol c and has the form d/d f (, ). The smbols c and k represent constants. d. d c e k 2. d d ( c e 2 ) In Problems 3 and 4, fill in the blank and then write this result as a linear second-order differential equation that is free of the smbols c and c 2 and has the form F(, ) 0. The smbols c, c 2, and k represent constants. d 2 3. d (c 2 cos k c 2 sin k) d 2 4. d (c 2 cosh k c 2 sinh k) In Problems and 6, compute and and then combine these derivatives with as a linear second-order differential equation that is free of the smbols c and c 2 and has the form F(,, ) 0. The smbols c and c 2 represent constants.. c e c 2 e 6. c e cos c 2 e sin CHAPTER in Review 29

12 In Problems 7 2, match each of the given differential equations with one or more of these solutions: (a) 0, (b) 2, (c) 2, (d) In Problems 3 and 4, determine b inspection at least one solution of the given differential equation ( 3) In Problems and 6, interpret each statement as a differential equation.. On the graph of f(), the slope of the tangent line at a point P(, ) is the square of the distance from P(, ) to the origin. 6. On the graph of f(), the rate at which the slope changes with respect to at a point P(, ) is the negative of the slope of the tangent line at P(, ). 7. (a) Give the domain of the function 2/3. (b) Give the largest interval I of definition over which 2/3 is a solution of the differential equation (a) Verif that the one-parameter famil c is an implicit solution of the differential equation (2 2) 2. (b) Find a member of the one-parameter famil in part (a) that satisfies the initial condition (0). (c) Use our result in part (b) to find an eplicit function f() that satisfies (0). Give the domain of f. Is f() a solution of the initial-value problem? If so, give its interval I of definition; if not, eplain. 9. Given that 2 is a solution of the DE 2. Find 0 and the largest interval I for which () is a solution of the IVP 9 2, ( 0 ). 20. Suppose that () denotes a solution of the initial-value problem 2 2, () and that () possesses at least a second derivative at. In some neighborhood of, use the DE to determine whether () is increasing or decreasing, and whether the graph () is concave up or concave down. 2. A differential equation ma possess more than one famil of solutions. (a) Plot different members of the families f () 2 c and f 2 () 2 c 2. (b) Verif that f () and f 2 () are two solutions of the nonlinear first-order differential equation ( ) (c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either famil of solutions in part (a). 22. What is the slope of the tangent line to the graph of the solution of 6! 3 that passes through (, 4)? In Problems 23 26, verif that the indicated function is an eplicit solution of the given differential equation. Give an interval of definition I for each solution cos 2 sin ; sin cos 24. sec ; sin (cos ) ln(cos ) ; sin(ln ) sec(ln ); cos(ln ) ln (cos(ln )) (ln ) sin(ln ) In Problems verif that the indicated epression is an implicit solution of the given differential equation. 27. d d 2; a d d b 2 2; ( 2 7) ( ) 3 ; ( ) 2 0; e In Problem 3 and 32, verif that the function defined b the definite integral is a particular solution of the given differential equation. In both problems, use the Leibniz formula for the derivative of an integral: d d# v() dv F(, t) dt F(, v()) d 2F(, u()) du d # v() 0 F(, t) dt. 0 u() 3. 9 f (); () 3# 0 f (t) sin 3( t) dt cos( sin u nu) du, 0 where n 0,, 2, [Hint: After ou have substituted,, and into the DE, then compute ( 2 n 2 ) 0; () p# p d ( cos u n) sin( sin u 2 nu) du and look around.] 33. The graph of a solution of a second-order initial-value problem d 2 /d 2 f (,, ), (2) 0, (2), is given in FIGURE.R.. Use the graph to estimate the values of 0 and. FIGURE.R. Graph for Problem 33 u() 30 CHAPTER Introduction to Differential Equations

13 34. A tank in the form of a right-circular clinder of radius 2 feet and height 0 feet is standing on end. If the tank is initiall full of water, and water leaks from a circular hole of radius 2 inch at its bottom, determine a differential equation for the height h of the water at time t. Ignore friction and contraction of water at the hole. 3. A uniform 0-foot-long heav rope is coiled loosel on the ground. As shown in FIGURE.R.2 one end of the rope is pulled verticall upward b means of a constant force of lb. The rope weighs lb/ft. Use Newton s second law in the form given in (7) in Eercises.3 to determine a differential equation for the height (t) of the end above ground level at time t. Assume that the positive direction is upward. lb upward force (t) FIGURE.R.2 Rope pulled upward in Problem 3 CHAPTER in Review 3