Math 42 Chapter 7 Practice Problems Set B
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1 Mth 42 Chpter 7 Prctice Problems Set B 1. Which of the following functions is solution of the differentil eqution dy dx = 4xy? () y = e 4x (c) y = e 2x2 (e) y = e 2x (g) y = 4e2x2 (b) y = 4x (d) y = 4x (f) y = 2x 2 (h) y = 2e 4x 2. Which of the following functions re solutions of the differentil eqution y + y = 6y? Show your work. () y = e 4x (c) y = e 2x (e) y = e 2x (g) y = 2 sin(2x) (i) y = 3e 3x (b) y = e 3x Direction Fields (d) y = 4e 2x2 (f) y = 2x 2 (h) y = 4e 4x (j) y = 3e 2x 3. Mtch ech of the slope fields below with exctly one of the differentil equtions. (The scles on the x- nd y-xes re the sme.) Also, provide enough explntion to show why no other mtches re possible. (i) y = xy + 1 (ii) y = sin x (iii) y = xe y (iv) y = y (v) y = sin y () (c) (b) (d)
2 4. The slope field for the differentil eqution y = 0.5(1 + y)(2 y) is shown below. (The scles on the x- nd y-xes re the sme, nd some of the vlues from 2 to 2 re mrked on the y-xis.) () For which regions re ll solution curves incresing? differentil eqution. Justify your nswer using the (b) For which regions do the solution curves tend towrd finite y-vlue s x? Justify your nswer using the differentil eqution. 5. A direction field is given in the picture below. Which of the following represents its differentil eqution? Explin why ech of the other differentil equtions is not represented by the direction field. () y = y x (b) y = y 2 x 2 (c) y = y + x (d) y = y 2 + x 2 (e) y = y x 2 (f) y = x y 2
3 Seprtion of Vribles 6. Solve the differentil eqution dy dx = x + sin x 3y Solve the initil vlue problem dy dx = 4 7y, y(0) = 3. Show ll of your work, with full mthemticl justifiction. 8. In this problem, we will solve the differentil eqution xy + 2y = cos(x 2 ), even though it is not seprble eqution. () Suppose y(x) stisfies the bove eqution (for x 0). z(x) = x 2 y(x) stisfies z = x cos(x 2 ). (b) Use seprtion of vribles to find ll solutions to z = x cos(x 2 ). (c) Solve the initil vlue problem xy + 2y = cos(x 2 ), y( π) = 0. (Hint: remember, the function x 2 y(x) is solution to prt (b).) Verify tht the new function 9. An eqution used to model the growth of niml tumors is given by y = y ln(y/b), where nd b re positive constnts. (This is known s the Gompertz eqution.) () Find ny equilibrium solutions of the Gompertz eqution. (b) If y(0) > 0, which one of the following vlues equls lim t y(t)? (i) 0 (ii) (iii) b (iv) 1 (v) 1 2 (vi) /2 (vii) b/2 (c) Solve the Gompertz eqution. Show ll of your work, nd be sure to find y s function of t. (d) Verify tht the nswer you obtined in prt (c) stisfies the Gompertz eqution. Show ll of your work. 10. A tnk is constructed in the shpe of cone, with vertex pointing down, hving height H nd bse rdius R. The vertex of the cone hs vlve which releses wter; the rte dv of decrese in the volume of wter in the tnk t time t is proportionl to the wter s height h(t) t time t. Let k be the constnt of proportionlity. () Find the differentil eqution stisfied by the height h(t). (Hint: when the wter in the cone hs height h, the volume of this wter is V = 1 3 π ( R H h) 2 h.) (b) Solve the bove eqution for h(t), subject to the initil condition tht the tnk is full t time t = 0. (c) How long does it tke for ll the wter to drin out?
4 Exponentil Growth nd Decy 11. A lef of lettuce from one of the dining res on cmpus contins pproximtely % s much crbon-14 s freshly cut lettuce lef. (Recll tht upon hrvesting, the C 14 in living object decys with fixed hlf-life; use 5730 yers for the vlue of the hlf-life.) () How long go ws the lettuce lef hrvested? (b) How much crbon-14 would one-week-old lef of lettuce contin? 12. A popultion growing t constnt reltive growth rte tkes 20 yers to triple. How long does it tke for the sme popultion to double? 13. In 1970, the Brown County groundhog popultion ws 100. By 1980, there were 900 groundhogs in Brown County. If the rte of popultion growth of these nimls is proportionl to the popultion size, how mny groundhogs might one expect to see in 1995? 14. The method of crbon-14 dting ws used to trce the successive emergence of the Hwiin islnds from the oldest, ui, to the youngest, Hwii. The islnd of Hwii is pproximtely 100, 000 yers old. Wht frction of its originl Crbon 14 does it contin? (Recll tht the hlf-life of crbon-14 is pproximtely 5730 yers.) The Logistic Eqution 15. A popultion P (t) grows ccording to the logistic eqution, with initil reltive growth rte 0.05 nd crrying cpcity 30,000. If the popultion is initilly equl to 5700, write down n initil vlue problem for the popultion. (You do not need to solve it.) 16. The growth of certin rbbit popultion is given by the logistic eqution with proportionlity constnt (i.e., initil reltive growth rte) 0.25 when time is mesured in months. Assume tht t time t = 0, the popultion is equl to 1 percent of its crrying cpcity. How mny months does it tke the popultion to climb to level of 50% of its crrying cpcity? Multi-Topicl Questions 17. In this problem, we will nlyze the differentil eqution dp = 1 ( 2 P 1 P ) () Suppose P (t) represents the number of mture lemons on tree t time t, where t is mesured in months. Explin the significnce of the term 6. (Hint: you might suggest n interprettion for this term.) (b) Find ny equilibrium solutions to the differentil eqution. (c) For which vlues of P is P (t) incresing? decresing? (d) Using the eqution, find the popultion for which the increse in the popultion is the most rpid. Be sure to include ll of the detils necessry for complete solution.
5 (e) Use the eqution to show wht hppens to P in the long run. Be sure to consider the cses for vrious initil vlues of P. (f) Solve the initil vlue problem dp = 1 2 P ( 1 P ) 6, P (0) = In mny species, if the popultion density flls below certin level, then the popultion my experience very low reproduction rtes, due to n inbility to find mtes. (This is clled n Allee effect see Animl Aggressions: A Study in Generl Sociology, by W. C. Allee, University of Chicgo Press, 1931.) This cn be modeled with n extension of the logistic eqution s follows. Let N = N(t) be the density of popultion t time t. Then dn = rn ( 1 N ) ( 1 ), N where r, nd re positive constnts. We ssume tht 0 < <, where is the crrying cpcity. The constnt is clled the threshold popultion size. () Find ll equilibrium solutions of the differentil eqution. (b) Which of the following best depicts the direction field of the bove eqution? I III II IV (c) If solution curve (other thn n equilibrium solution) pproches n equilibrium vlue s t, such limiting vlue is clled stble equilibrium. Determine which (if ny) of the equilibrium solutions from prt () re stble. Give full mthemticl justifiction of your nswers, using the differentil eqution.
6 Systems of Differentil Equtions 19. The differentil equtions shown below form predtor-prey system, with popultions x(t) nd y(t) nd positive constnts, b, c, d. dx = x bxy dy = cy + dxy () Which popultion is the predtor nd which is the prey? Explin/justify your nswer using the differentil equtions. (b) Which constnt should be lrger, b or d? Give s thorough n rgument s possible to support your nswer. 20. Assume tht we hve n environment populted by five species, whose popultion sizes re given by the functions v(t), w(t), x(t), y(t), nd z(t). Suppose tht x, y, nd z stisfy the system of differentil equtions: x = 0.1x 0.005xz Furthermore, v nd w stisfy the system y = 0.004y yz z = 0.03z xz yz v = 0.04v vw w = 0.1w vw Discuss which species re predtors upon which species, which species cooperte with which species, nd/or which species compete with which species. 21. Suppose we hve n environment populted by rbbits nd wolves, nd tht we use the usul system of differentil equtions to model their popultions: where k = 2.4, = 0.4, r = 10, nd b = 0.2. dr = kr RW dw = rw + brw () Find ll equilibrium solutions (R, W ) where both the rbbit nd wolf popultions re constnt. (b) Suppose tht sudden nturl disster instntly reduces ech of the rbbit nd wolf popultions to 50% of their equilibrium sizes. Indicte the rte nd direction of chnge tht ech popultion will initilly experience fter this point in time. (c) Suppose tht the wolf popultion drops to zero t point where the rbbit popultion is 60. Find n explicit formul for the rbbit popultion s function of time, s indicted by this system of equtions.
7 22. Two compnies shre the mrket for new technology. Let x(t) be the net worth of one compny (in millions of dollrs) nd let y(t) be the net worth of the other compny (in millions of dollrs), t time t months. Suppose x nd y stisfy the differentil equtions x = x xy y = 2y xy () Find the equilibrium solutions for this system of differentil equtions. (b) Find n expression for dy dx. (c) (For this nd prt (d), use the system s slope field, provided below. As usul, x is plotted on the horizontl xis, nd y on the verticl. Note tht rrows hve not been drwn on the slope lines.) Explin wht hppens to the net worths of the compnies in the long run if x(0) = 4 nd y(0) = 3/2. Also, drw trjectory (tht is, solution curve) on the slope field below tht strts t x(0) = 4 nd y(0) = 3/2. Include direction rrow on your trjectory (d) Explin wht hppens in the long run if initilly x = 4 nd y = 3/2, but becuse of venture cpitlist investments, very soon fter t = 0, y is incresed by 2. Also, drw trjectory on the slope field below tht shows this scenrio, strting t x(0) = 4 nd y(0) = 3/2. Include direction rrow on your trjectory
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