problems_1-1and1-2_20110512_s11

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Unformatted text preview: , we utlljze: s or functio Jemher of " at be obtain ion is called“. solutions particular 3 ar solutio choice of th val I can I es of the 3 general, 0 . )lu'tions iations ne c1 + {325 (- +1nc2xisasov ). Are q an SECTION 1 .1 Basic Definitions and Terminology 9 1' 4'Sha1'1 11:19.03:rélativelnsimple iéstrictiiim (011.111? CQB‘ffisiPntSi-Withthese" f =‘*=resiri,¢tions19n6 fianelivaii he assured net. .dnly‘thatj 3. ifsomtidn.-do¢s.exis'.t -. 'aii interyal hotelsothat'a family or solutions'indeed yields an possible ' ;.2' _ ‘ Anothe 'fao't deserives mention at this time. Nonlineareqiiations,.f§vith 7 : tame. emotionof-simemsmaeraqueous,areshinydimcultarimpbs- fj_.$ibfeto solue inj'teijrris of elem-enta'ryifunc'tiohsisueh as aigebraio -'_'fiirietioris,"eiiponential andlogerithmio _'fiunotions,:and trigonometric and ._ .- inverse trigonometrie functions.._Furthermo}ce, if-Wehafipeii tohave a__fan__1_—-_3_ solutions for 'a nonlinear equation, 'iti'isi'notobvious-when this fang—ff ‘ {fly-Cofistifutesfa'aehetalsomtiéhiOn.-a"i>ra¢¢ical.1¢¥cii'ihéni'flte 'des’igiia— '- i ,'.t'ion “generai solution”__i_s applied only-to linear differential equations. r}. -2' -. ' . .1.- .'éx:aéiisis i151 .; Answers to odd-numbered problem begin on page A-I. In Problems 1—10 state whether the given differential equations are linear or nonlinear. Give the order of each equation. d3 d 4 1. (1~x)y”H4xy' +5y== cosx 2 tel—2(1) +y=0 ‘ dx3 ' dx 3. yy’+2y=1+:c2 4. xzdy+(y—xyh—xe‘)dx20 d2 5. x3544) — x232" + 4xy’ — 3y = 0 6. 817}; + 9y = siny dy (rial): dzr k . _ = + —— 8. — = ~— 7 dx I ah:2 dig r2 9. (sin x)y”’ — (cos x)y’ = 2 10. (1 H y2)dx + xdy = (3 in Problems Iim40 verify that the indicated function is a solution of the given differential equation. Where appropriate, 6; and 62 denote constants. 11.2y’+y=0; y=e#"2 12.y’+4y=32; y=8 d 13.;g—2yz'e3"; y=«I33’W-‘ltlezI dy 6 6—20: 14.E+20y=24; y=§“§8 15. y’m25+y2; y=5tan5x d 16.39%: g; y=(\/E+c1)2,x>0,c1>0 17. y’ +y : sinx; y = %sin. —%cosx + 109:—Jr 18. ny dx + (x2 + 2y) dy : 0; xzy + yz __'_Acl 19.3.t2dy + ny dx = 0; y = _ :5 20. (y’)3+xy’ =y; y=x+1 21- y = 2x)” + WT; y: = 530‘ + ici) CHAPTER 1 INTRODUCTION TO DIFFERENTIAL £QUATIONS 21Y=2VM;y=me 23.y’—$y=1;y=xlnx,x>0 dP acle‘" m~=P ~bP'P=w¥~AA 24 dt . (a )’ 1 + bale“ dX 2 — X 3 I _ —_~ _ _ - 1 z . 25 d? (2 X)(1 X), n 1 fl X r 26. y' + ny = 1; y = e”x2J:e‘2dt + 616")“: 27. (x2 + yz) dx + ()1:2 — xy)dy = 0; c1(x + y)2 = xefl" 28. y" ~1- y' — 12y = O; y = ole” + age—4" 29. y" — 6y' ‘+ 13y = O; y = 63" cost 39.£X5~4%+4y=0; y‘=ez“-£-.xezJr .. 31. y" = y; y = coshx + sinh x _ 3 32. y” + 25y = 0; y 2 goes it 33. y" + (y’)2 = 0; y : lnlx + all + C2 34. y” + y = tan x; y = ~c0sxln(secx + tan x) 35. xj—E; + 2% = 0; y = c1 + czx‘1 36. x2)!” — xy’ + 2y = 0; y = x cos(ln x),x > 0 37. x2)!” - 3xy’ + 4y = 0; y = x2 + x2111 16,); > 0 38. y'” H y” l 9y’ H 9y = O; y = 61 sin 3x + £2 cos 3); + 43x 39. y’” w 3y” + 3y’ m y = 0; y = xze" d3y d2 t d . . 4t}. XBE'F 2x2? ~x§ +y = 12x2; y= c1x+c2xh1x+4x2,x>€l In Problems 41 and 42 verify that the indicated piecewise-defined function is? solution of the given differential equation. —x2, x<0 v ; 41.xy’__2y:0; y:{x2 x20 0 x<0 4a 'Zzgw r: ’ 0) M ) {fl x20 43. Verify that a one—parameter family of solutions for y:xy’ +0202 is y=cx+cz. Determine a value of it such that y 2 lot2 is a singular solution of the di ferential equation. 44. Verify that a one-parameter family of solutions for y=xy’+\11+(y')2 is y=cx+V1+c2. *2 Show that the relation x2 + y2 : 1 defines a singular solution of the equl tion on the interval (—1, 1). motion SECTION 1 .2 Some Mathematical Models 45. A one-parameter family of soiutions for 1 + cez“ —Zx. y’=y2—1 is y= l—ce By inspection,* determine a singular solution of the differential equation. 46. On page 6 we saw that y r V 4 H x2 and y = m V 4 m x2 are solutions of dy/dx = —x/ y on the interval (—2, 2). Explain why ={ Vii—x2, —2<x<0 y "V4—x2, 05x<2 is not a solution of the differential equation on the interval. In Problems 47 and 48 find values of m so that y = e” is a solution of each dif- ferential equation. 47. y” — Sy’ + 6y = 0 48. y” + 10y' + 25y = 0 In Problems 49 and 50 find values of In so that y = x’” is a solution of each dif- ferential equation. 49. xzy" — y = 0 SI}. xzy” + 6xy’ + 4y = 0 51. Show that y} : x? and y; = x3 are both solutions of x2)?” - 4xy’ + 6y 2 0. Are the constant multiples c1 yl and ngz, with Cl and C2 arbitrary, also solu— tions? Is the Sum yl + yz a solution? 52. Show that y1 = 2x + 2 and ya = 1162/2 are both solutions of y = xy’_+ to»)? Are the constant muitiples c; yl and c2512, with Cl and c2 arbitrary, also solu- tions? Is the sum y; + y; a solution? 53. By inspection determine, if possible, a real solution of the given differential equation. ' dy dx Liz dx +M=1 (b) +Iyt+'1 =0 (c) In science, engineering, economics, and even psychology, we often wish to de- scribe or model the behavior of some system or phenomenon in mathematical terms. This descriptions starts with (1) identifying the variables that are responsible for changing the system and v (if) r a set of reasonable assumptions about the system. * Translated, this means take a good guess and see if it works. 22 CHAPTER 1 Figure l.l4 Figure I.I5 INTRODUCTION TO DIFFERENTIAL EQUATlONS ' - a ms dt r where r is the annual rate of interest. This mathematical description is a ogous to the population growth of Example 10. The rate of growth is larg when the amount of money present in the account is also large. Transl geometrically, this means the tangent line is steep when S is largef Figure 1.14. The definition of a derivative provides an interesting derivation of ( Suppose SQ) is the amount of money accrued in a savings account after t y when the annual rate of interest r is cempounded continuously. If h denotes increment in time, then the interest obtained in the time span (t + it) — tis the difference in the amounts accrued: So + h) u 30‘). Since interest is given by (rate) >< (time) x (principal), we can approximate interest earned in this same time period by either rhS(t) or rhS(r+h). Intuitiver we see that rhS(t) and rhS(t + h) are lower and upper bou: respectively, for the actual interest (27); that is, rhS(t) s S(r + h) — S(r) s rhS(r + h) ‘ 80‘ + h) H S0) 12 Taking the limit of (28) as h —> 0, we get S(t + h) u S(t) or rS(f) s: s rS(t + h). rS(t) s h _<_: rS(t), and so it must follow that I S(t+h)~"S(t)_ dS_ hmmW — 1S6) or E}- — rS. Answers to odd-numbered problems begin on page A-J. In Problems 1—22 derive the differential equation(s) describing the given 1’1 ical situation. 1. Under some circumstances a falling body B of mass m, such as the skydi shown in Figure 1.15, encounters air resistance proportional to its instatk neous velocity v. Use Newton’s second law to find the differential equa for the velocity v of the body at time t. Recall that acceleration a 2 dv/dr. Assume in this case that the positive direction is downwat C 1-9, B W 2. What is the differential equation for thevelocity v of a body _ falling vertically downward through a medium (such as water) that off? resistance proportionai to the square of its instantaneous velocity? ASS“ the positive direction is downward. ' he given 3. 10. SECTION 1.2 Some Mathematical Models 23 By Newton’s universal law of gravitation the free-fall acceleration a of a body, such as the satellite shown in Figure 1.16, falling a great distance to the surface is not the constant g. Rather, the acceleration a is inversely pro- portional to the square of the distance r from the center of the earth, a = k/r 2, where k is the constant of proportionality. (3) Use the fact that at the surface of the earth r = R and a = g to deter- mine the constant of proportionality k. (b) Use Newton’s second law and part (a) to find a differential equation for the distance r. (c) Use the chain rule in the form hieraa dt2 (12‘ dr air to express the differential equation in part (b) as a differential equation involving v and dv/dr. (a) Use part (b) of Problem 3 to find the difierential equation for r if the re sistance to the falling satellite is proportional to its instantaneous velocity. (1)) Near the surface of the earth, use the approximation R w r to show that the differential equation in part (a) reduces to the equation derived in Problem 1. A series circuit contains a resistor and an inductor as shown in Figure 1.17. Determine the differential equation for the current Kt) if the resistance is R, the inductance is L, and the impressed voltage is E(!). A series circuit contains a resistor and a capacitor as shown in Figure 1.18. Determine the differential equation for the charge (30‘) on the capacitor if the resistance is R, the capacitance is C, and the impressed voltage is E(t). Suppose a tank is discharging water through a circular orifice of cross- sectional area A0 at its bottom. It has been shown experimentally that when friction at the orifice is taken into consideration, the volume of water leav- ing the tank per second is approximately 0.6140 V 2gb . Find the differential equation for the height h of water at time t for the cubical tank in Figure 1.19. The radius of the orifice is 2 in. and g = 32 ft/sz. A tank in the form of a right circular cylinder of radius 2 ft and height 10 ft is standing on end. The tank is initially full of water, and water leaks from a circular hole of radius % in. at its bottom. Use the information in Problem 7 to obtain the differential equation for the height h of the water at time t. A water tank has the shape of a hemisphere with radius 5 ft. Water leaks out of a circular hole of radius 1 in. at its flat bottom. Use the information in Problem 7 to obtain the differential equation for the height h of the water at time I. The rate at which a radioactive substance decays is proportional to the amount A(t) of the substance remaining at time 2‘. Determine the differen- : tial equation for the amount A(:).' 11.‘ A drug is infused into a. patient’s bloodstream at a constant rate of r grams per second. Simultaneously, the drug is removed at a rate proportional to the amount x0) of the drug present at time t. Determine the differential equation governing the amount x0). ...
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This note was uploaded on 12/30/2011 for the course MAP 2302 taught by Professor Jones during the Summer '11 term at Tallahassee Community College.

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