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Unformatted text preview: 404 I CHAPTER7 APPLICATIONS OF INTEGRATION m EXAMPLE 6 (a) Sketch the direction field for the differential equation y’ = x2 + yz — 1. (b) Use part (a) to sketch the solution curve that passes thrOugh the origin. SOLUTION I (a) We start by computing the slope at several points in the following chart: ' [5, Fin x J—2—1012—2-1012...am 1 6, Fin y‘ 0000011111...“ ‘y'=x2+y2—l 3 O -1 0 3 4 l 0 l 4 .100 (b) Now we draw short line segments with these slopes at these points. The result is the . direction field shown in Figure 6. (C) - (b) We start at the origin and move to the right in the direction of the line segment (which has slope — 1). We continue to draw the solution curve so that it moves para] 8. 301V lel to the nearby line segments. The resulting solution curve is shown in Figure 7. me" Returning to the origin, we draw the solution curve to the left as well. , curv ‘fSolv The more line segments we draw in a direction field, the clearer the piclur y(0) becomes. Of course, it’s tedious to compute slopes and draw line segments for a hug impl: number of points by hand, but computers are well suited for this task. Figure 8 sho“ a more detailed, computer-drawn direction field for the differential equation 1' ' 31:1“ . . m FIGURE 7 . ‘ Example 6. It enables us to draw, With reasonable accuracy, the solution curves SI'10\\‘ impli in Figure 9 with y-intercepts —2, - 1, 0, 1, and 2. const Module 7.6 shows direction 44 I . (. elds and solution curves for a be] d ' variety of differential equations. ‘ e I' ; y' = l y' = FIGURE 8 FIGURE 9 I p 1.6 ] EXERCISES l-8 I Solve the differential equation. 9—” . Find the solution of the differential equation that Sill" fies the given initial condition. I 9.11.:2’. 2 _d_y_= fl dx x dx ey 3. (x2 + l)y’ =xy 4. y’ =y2 sinx V dy e"sin20 5.1+ta '=2+1 6.——= ( my x d0 ysec6 II. x cosx = (2y + e3")y’, y(0) = 0 du pm dz 1,, 7.—d—=2+2u+t+tu $§§§—£j—+g’—=O W t t l2.gt—=\/P_t, P(1)=2 . I I I I 15- Find. anééuatioaotflia only? thatsatisffieé oldie My and whose y-intercept is 7." ' ' ’ ' - [5, Find an equation of the curve that passes through the point ('1, 1) and Whose slopeat (x, y) is yZ/x3. ' ' ' . 11,.(a) Solve the differen ‘al equation y’ = th/l — y. (b) Solve the initial—value prOblem 'y’ = 2xt/l — yz, y(0) = 0, and graph the solution. . . : )Does the initial—value‘problem'y’ = 2x1/1 — yz, I y(0) = 2, have a solution? Explain. . 'he result is the : line segment tit moves paraI 1 in Figure 7. I. lve" the equation e"y’ + cos x = O and graph several members of the family of solutions. How does the solution ,‘ciurve change as the constant C varies? Ive the initial—value problem y’ = (sin x)/ sin y, arer the piCtlHe y(0) = 77/ 2, and graph the solution (if your CAS does ments for a hue! £33313:th “ , ' ' t V '4‘? + '1/(ye’J'land-grapfi several l 3 ‘ ‘ ‘ ouriCAS'do'es' ‘ .on curves 5110“ y ' the solution ionrve, cha'ng‘e‘as the Match the differential equationvwith its directionfield IV). Give reasons for your ansWer. ,y 1' I . 23- y’ =xt2 —y) 24. y’ = sinx siny / ///// / ////// / / / ecluation'that 5“ { \ \\\\\ . \ \ \\\\\ \\\\\\\ \\\\\\\ \\\\\\\\ \\\\\\\\ \\\\\\\\ \\\\\‘\\\ i (A. I ./ {Qii /.//.////' /' / ////// ' ///// //// /// // i: ’2////// // ~ .‘ SECTI'ON“7.6 ~ DIFFERENTIAL EQUATIONS I 405" 25-26 I Refer to the direction fields invExercisesZ'I—Zél. j 25.‘ ' Use field III-to sketch the graphs of the solutions that satisfy, ' ' - the given initial conditions. _‘ ‘ , (a) y(0) = 1 . - 0’) ya» = 2 (0) VIC) ‘= “1 26. ‘Use field IV tosketch the graphs; of the solutions that satisfythe giveninitial conditions. ‘ (a) y(0) = —1 (b) y(0) = 0 (C) N» = I 27—28 - Sketch a direction field for the differential equation. Then use it to sketch three solution curves. 27.y’=1+y 23_ y'=x2—ly2 I I I I . I I I I I I I I I 29-32 - my’=y—2x, (1,0) 30. y’=l—xy, (0,0) (1, 0) £3? y’ = y + xy, (0.1) 32. y’ = x — xy, % Psychologists interested in learning theory study learning I , V V ' curves. A learning curve is the graph of a function P(t), the ' pe _or1nance Of someone learning a skill as a function of the' training time t. The derivative dP/dt represents the rate' at which performance improves. p ‘ . . . (a) If M is the maximum level of performance of which the learner is capable, explain why the differential equation dP —=k(M—P) dt k a positive constant is a reasonable model for Ieaming. (b) Solve the differential equation in part (a) to find an expression for P(t). What is the limit of this expression? 34. A sphere with radius 1 m has temperature 15°C. It lies inside a concentric sphere with radius 2 m and tempera ture 25°C. The temperature T(r)‘ at a' distance r from the common center of the spheres satisfies the differential: ‘ equation. ' V ‘ ' ' " ’ V If we let S = dT/dr, then S satisfies a first—order differential equation. Solve it to findlan expression for the temperature , . . y , T(r) between the spheres. " ' ' Q A glucose solution is administered intravenouslyiiiito the -. , V -' bloodstream at a constant rate r. As’ the glucose is added, it ,_ isconverted intoother s'ubstances'and removed from the ” bloodstream at a rate that is proportional to the~concen.—. .. tration at that time. Thus a model for the concentration 3406]“- CHAPTER?! IAPPLICATIONS OF-lNTEGRATION C é. C(t)‘.of the glucose solution in the bloodstream is d: , kc. wherek is a positive constant. (a) Suppose that the concentration at time t = 0 is Co. Determine the concentration at any time t by solving the differential equation (b) Assuming that Co < r/ k, find limHm C0) and interpret your answer. 36. A certain small country has $10 billion in paper currency in circulation, and each day $50 million comes into the coun- ' try’s banks. The government decides to introduce new cur— rency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x = x(t) denote the amount of new currency in circulation at time t, with x(0) = 0. (a) Formulate a mathematical model in the form of an initial—value problem that represents the “flow” of the new currency into circulation. ' (b) Solve the initial-value problem found in part (a). (c) How long will it take for the new bills to account for 1.. -’~ 90% ofthe currency in circulation? 31. Write the solution of the logistic initial-value problem V i: - dt 0.00008P(1000 — P) P(0) = 100 and-useit to find the population sizes P(40) and P(80). At what time does the population reach 900? The Pacific halibut fishery has been modeled bythe differ— 38. ential equation _, dy —- = k M — dt y( y) where 370) is the biomass (the total mass of the members of the population) in kilograms at time I (measured in years), the carrying capacity is estimated to be M = 8 X 107 kg, andk= 8.875-X 10'9 per year; V. , . .' p (a) If y(0) = 2 X 107 kg; find the bi'omassa year later. A ' (b) How long will it take the biomass to reach 4 X 107 kg? 33% One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. (a) Write a differential equation that is satisfied by y. (b) Solve the differential equation. V (c) A small town has 1000 inhabitants. At 8 AM, 80 people have heard a rumor. By noon half the town has heard it. At what time will.90% of the population have heard the III 40. Biologists stocked a lake with 400 fish of one species and 43. An 01 .v . h 42. 43. 44. 45. 46. if? When a raindrop falls, it increases in size and so it estimated the species’ carrying capacity in the lake to be 10,000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. \ (b) How long will it take for the population to increase to 5000? s (a) Show that if y satisfies the logistic equation (7), then d2 3 = 18th — y)(M — 2y) (b) Deduce that a population grows fastest when it reache) half its carrying capacity. For a fixed value of M (say M = 10), the family of legism functions given by Equation 8 depends on the initial value yo and the proportionality constant k. Graph several mem- bers of this family. How does the graph change when yo varies? How does it change when k varies? A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/ min. The ' solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank (a) after t minutes and (b) after 20 minutes? The air in a room with volume 180 m3 contains 0.15% car- bon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 m3/min and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the room as a function of time. What happens in the long run? A vat with 500 gallons of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the val 2115 rate of 5 gal / min and the mixture is pumped out at the w rate. What is the percentage of alcohol after an hour? A tank contains l000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a ralctl 5 L/ min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/ min. The solutI : kept thoroughly mixed and drains from the tank at a rate 15 L/ min. How much salt is in the tank (a) after tminlll and (b) after one hour? time t is a function of t, m(t). The rate of growth of the " is km(t) for some positive constant k. When we apply , ton’s Law of Motion to the raindrop, we get (mv)' =9“ where 11 is the velocity of the raindrop (directed doWn and g is the acceleration due to gravity. The terminal " E ity of the raindrop is limp... v(t). Find an expressiml,f terminal velocity in terms of g and k. Slllllss.‘ L" medit tion c where tion 0: of a b1 (a) Su ...
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