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Solution Manual For Advanced Engineering Mathematics, 2nd Edition

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    Section 1.2 1 Chapter 1 Section 1.2 1. (b) Fun ondun 1 2Rma eK F#9Mm2K so NO. Mp? 2(3RmXY360K) doco = 9/unm2x (hee 2RUANCOK = MM 2X) Ao Yew, Ms: 2(e*ye™)# 9 max Ao No. 2 2 2 2 eZ : 4 hy miranm -1= ~2RE™ [Feat + AEX € rahe” [TET dt =0 ody A BE Sea, on rid, No Ty on Mr c= -2xe LE Var + £7 er axe” [ak <1 deo 20 fn Ql chocsa 4 a, ro He a y Maw, Maz ac $ Uy t A, App = (2 425 ring, imbs puwhez =0 of edie (ay azb=c / =0 po Na Amaz dnb Awchez =0 "Wt den io a ude 1 Ct=ai+LY)! 5. (b) mie3mt = xe +3E2F = 2% (arzeM). The & fain Ao nd 0 fon amy 20, Lk dmg fn afl x. Gnd fon Fhe neand fan Jo be 0 a1 al, x Sugg dh EM Jo a conitunt and IE, in Tarn nasi That A=0. Buty 20 thin A+3€M =0+3#0. Tha, mo such adutiois © u'-3y'r2y =(N-snt+2)e™ zo 4 Xe3ne2 20, Le, JAI 2. Thao, €* amd EH ane pdudimo. we 6. (5) m'-y-x* = (2+ Aoinh xt Beh )-(-Kro+ Asche + Bopht)~x* doto= 0, qo=-z=-2tB amd 40202 A gre AsB=0, so Mi) = -X*2. 7. (0) Nedomian due To the An Tor d) Nonbwaon due To the Ltp() Fn, ® Nedinean due To The mm Lown. OH) ARs, Jonson. 8 wx C, sme ytd Section 1.3 3.@ Since AW= wars pas, te neg That a = juds/dx =p Aig. 2 (10) and. (11), Tentr= A . Jo ng to ac Tamg=pf diya +B) Two A and dfx quia y's Cig. Chapter 2 Section 2.2 . - py 4x 2.09 wtdn=8, so a) gra i= €5 (oan + C)= E(B 4 C) = 2¢Ce™. On, indigrating Fodor mdhed, cmorden of [Heo = 08. Fou ohlrdon k= (ayy = ana’ kt herd, T'=4T 40, few (7), oth e¥®. Thu, (@y)'=3e™, no e¥y= (“get d+ C on

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    Section 2.2 2 _ Fax. . moo= 2+ Ce 0 od (dt ©° y= gimme (Je Han c) = '%E "(fe Je crt C) =e (fef (eP%.) CE Gap tC) = ERIE ([etMPEl yc) = iz (felemxjax+C). Recall dha the Tama ain the ODE do dejentd by coy =3M/2< X<-TY2 Tp <X<TV/2, Wok X< 32 ye He. Om <x M2, frm, cox>0 ro lax = cox omd 400 = g5z (6pinx + C). Om m2 <x<3M/2, fou at, epE<O AO [CK] = - Cox amd ex) = (f-eoxdn + c) = go (6pinx = C), and room, AE on A dhe Bld. primal Fhe shedion soy 100) = gr (Git + “A7) I, Aso am rr] endund [&) Mix) = e (fef 1% gay +C) = esnH( fe? 1x1 29+ C) = pe ( [11 xFdot C) = L(fxtec)= £ + < fon D<x<D OL fen —~0<X<O. -3.(b) Mp0 = AEH no neck mon = Aue ™. Than (Re upé™)rupe®=g guia A'= 82%, A= fee ax + C = 26+ C, so Mz (2e%+ C)E* = 24 CEH, ao 4m 20). 5.) me) = 20+ Gx mh=2 = 2+ C gro C70 a go = 225 on - 0<x< 20, © fm= 25+ Cl. M@)=2=8+C2 oS - Goto C=-l2 26 H(X)= 2-12 on 0<X< 20. IR wo 6. (21) opurte. gpranal J Ws oy amas [ Rize (B) 40121 = 0+1+ 04 ae chore C=0. * fF Hoe, w= % +1. (Note: My unstiad : y YO pgs PF thn the Ao mo ad) 5 . on ~0<X< 20. Vert I [2 (© HEDFIZ-L +14 C guide Cd; po oq 4 ME) = Erle gh on -0<x<O. Z| T7..(5) Conandin. 2.2 X09) nathan Than a : 9 TH Y=). Thm &= exta® oy fed gowns, wp = <I (e iua) NI B00) Sh be mght fo ly Ho 00055, le J : 10<9<¢ part Sa othe diff, ming dhe elo) ae Phosspolialt (2+ 2ax-n)'s, Lol, aati (1203), S777] = -i--, 97d = [40,40] , lupe = 0.01, axrewo = LINE); 7 Aapage mado sedamesd a Phy T : re

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    Section 2.2 3 Rendon, the pdm, cure trough, [2,1] smoith £0 we wned, the edd ipial dion AMipego, = 001. (Fo futhen SE EAU command Ace the dedi) i} Row rene ah and peeking ol dhe ODE ) newaler dhe. pomp Find 55], #4 ic pores [TTT TTT ad binder Toth night, setally eppreschons =2%. Mi ar & harlot (-n'2)'2, leu], x= 2.2, it i / i £10,013, grid = [40,40], Alupruge = 0.04,=-3..3, 1 | omens UNE); goo dhe A ww MII Mev ot dhe nh. We done dhe aligned ZH 1 Cows YZ 443 and yy =-A3. rr i A Ao] =-3.. ee = { Grr rer mh pg z= HARE = LINE); gue He petra sham bilo To swede noflineenio ’ boast yt To 0.01 dnt dhe ep pis NEAT A FTW FAR.) o rtical Taegu of m=-i (a0 cam ’ abso prin fromthe ODE) amd dhim indo To dhe LE, an toheck cane. a ponshl -nnbisd . Fale “pTo dhe pont 4 rotieadl Fagin , dbs” 1 Cunt. NOTE: Y et mat pits die (nd didewned wndill Snes 4), a Roun , tom ample fom, dhe Aectind wsmz 549. Nak, dhe commando. : and ampuc tpl LRA (3-384 =%"3 +9, X= ~3..0, M=-3. isms = 510); eA ’ i= =| (9) o'= €%y. phospotiod (orp-2)xn, T,41, 2=0.20, {[0,101}, gd = [4940], gin =0..40, Aipogr = 0.04, anews = LINE); © the portrad sho on nent page. B phason ke Co mg, x41, x=0..00, Let eo ih, grid = [4040], pails = 0.04, annewd = LINE); duro dhe pened hn on maxt Page.

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    Section 2.2 ' 4. : (4) Cotionmsd : ) cndimued : i —— Eos ag : | | —— 5 Tn NN ———— a —— fe —————— IA _— J; iE lon. 222. sige, SE Y=0. Exact wntigral Cure ant = NTT (120,21, £2,...) 9.0) + pyzayt, ar=yT (no). W's (=m e"’ po 4'= f'n) amd the ODE ak of P= gy’ on, dung by v's (-n)pnr=@ in). - (2. 2dx 10.(6) n=2, bo N% Zr =-%? = € [fon (of% CR) dat c) “(fest C) == Ka Gy) po y= Phe (wh AzSC). 1 I", ER I+, CRA gate Vie (Te i¥+ (V+ £) +. Uaomg Yip Fer to bane Hove RCAC Js ? Wo p+ dm = -p- ME ou 12. (b) HE +1 ag p=hg=-x, =| amd. (11.3) Ao a+ xu =-l, u= (fey dat C) ou ut) = enti C-f*et?ar) pay. Thue, (12) gua ma)= x + e“”/(C- fe" at). (©) Fed Yi = axb= x*, $) Une Yn=1t Yy=2 th) Yi=22Vw=0 13.00) (13.3) a> lr =0, X+12x=0, wp = Ce SHFEMp 40 the prendre adTan in Up)= cep, Mp) = Xlpr FY) 1: AL = Ce (+3). . 5. 5 @ Pulling (134) Mo (13) goer xt §(6)= x FP) + 5(P) sohich so packed 4 p=codant =p, power, $RY=F,- a 4. (b) Stpy=p wm (132) gute O= [%f'(p)+ 5 pld 5 od By petifd be p= milan = C [Amer (11) goo (42)] Ly XE PY + 5p) =0. Smee Fp) 2 =p, dhe Latin gio x= gp) 1x : and (194) Gores N= Xp+ gp) = ~P§R + 4p) ©) Indho cane Ap =-pt a0 ¥ goo x= 2p, y= 2p-g= E- Dn dis cane wt salle Jo whomamade, p Adin (ho two o amd olla M= XY. 9

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    Section 2.3 | 5 A) The ponds so fhatithe Clans sqraFion \ | y (41) admits Lith dhe fomdy dt-boe / Addie (142) amd dhe adddidaad melition \ NY (143). Gorinadly Re antiga . Curwe Guren = AN pr. all 1.3) 10 am. ronrllope {d+ Nt ©), wo dhe paella m=, «0 NN : diopleud ct nught, Goone dhe freak - KX N\ _ fm, Migs which do wn shang cone “ h Fark mth Daan. sion aypoiy= 400, pdafins 4 ackih ane unigue (pudyct fo coded JOE on pix and 4005 pee Thacumn 12.2.1, pg 26) For tamepfle comin the hin) hon gh dn old prt 3,0). The sebtion come FHoeugh-that pod flo dhe % aio To x=-.Mo sgh follos dhe x ere ® fe en ybebod 2 Jneenio Tung Io dhe sdiFim came n=. Of dh pond I “Dao a chace? JX cam crfinie abogdte Kodo X=+0D NX camden ve Fhe “i (rm) at dhe Abveught Ee GE Er ESET La TAL Ig To Xx =+00. Two snch > are shoo dds by the Many Lino. . XA y / 2 > / ~ A A NA py yz E— a BY - ALL : = Xa Section 2.3 (FRc, t [TR 2.) it)=e€ bt “(f et EQ rsd) 3 LE, than ED)=0 on the signal, no itt)= & (041) ie, LS thon R . kt RT/L . (tL), |. REL ie (EH Bis Bee) wn 7 Stiady ST: dct)» Fogo gE — Addin . ee DDT RY i 3 Lo Vii grea , tl x % * ; ~Rt/L N 4: ict) = Seoks[e iQ cost)]. To chamae ¥ from Tun Trmo 15 one,

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    i . Section 2.3 6 Co wie Ramt-4) = Almotcoad-aind ail). Shut 7 comparing wnth ) Arc =0l Diwidurrg Guro Fad = WL/IR 1 P=tan! wL/K), amd, Rood =R | oyuaniog’and eddoig auto f= R00" 20 A= {RELY Thue, daty= El GRE, Eo al ot-d). Ror LY ARF (Ly 6. mity=m,€*% 40 g=106 0% Po ~ote=Lno, k= 000372 20 me) = |o@0-00372L . 2=10€ poli go t=4320 wp, pp 4 0110 eon guia X= 12379 pro. 7 mE) =me” . os, =e gece Ak = 0.00388. Thm, 05M, = wl, 0.003887 gwen T= 2094 dane. 12. @) mar'=mg-cnr; v0)=0. Thon 4 Snr =g (fnclrondin Lan Ln.) #0 wityz em ([efettimg urs £1) = eH (gfectimar if) = 1g Agim, Thon, N10) 20=19 +A guico A=— mile amd i § Nr) = BG (1- etm). Ao treo, AE)» 14 = Tomnal ray . (kb) mlz mg-cr 4 mow a. Radesdy squation’ (ne Exercise 11 on Sie 22) with. %, changed 54,0 and pARIE= Sm, 44) =0, t)= 4. Oloowing the protien, oeish {marc , change ApoB alle Secondung To w(t) =fmafe + Hs . Then the ODE Lecomen 0-4 =9- (I +k) oo =f R(T) on Woah od -N= J > == ne)= “+g +A TTL Run, w10)=0= BF Hil Fro HO) pe = 2 +f go R=-4f55 Fadl a3 vi “ — - = - amd {Et RE B8(- from) Toma wdlesity so {PGE 13. This problem is worked in the Answers to Selected Exercises. Here, we just wish to mention that to help the student feel more comfortable about the physical process of light extinction it might be useful to note the gradual extinction of light as we proceed deeper and deeper into the ocean. 14. NOTE: This problem is nice for use in class or lecture, especially in view of its environmental interest. Later on it will also make a nice example for the application of the Fourier transform, especially if the source is modeled as Q times a delta function at x = 0. The solution is given in the Answers to Selected Exercises, so here we will just give sketches : of the results and (for possible class discussion) give a brief formal derivation of the governing ODE. } (@) c (b) c | “7 Q/eA RC - x L vo

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    : Section 2.4 7 To duwe The Gavewang ODE od a masa badanet for om J nection hc rin, Lh 8d stad, Fuk Jane of ilficon pst Ut Sho 4 race, (4 phd.) ano dhe “urindgur” 4 wea Al x proportional He asua And dhe comcvdrotion anadunt cl) (rmiso Lrteanmne The ax and . bt from Hugh coneandnation fo lowr emcantraTiom, po €100>0 ll cause a flu, ’ pws, othe Aft ard cl) <0 will caune a Flu To the GMC) wunth a Att TT] i NES pr pe tds. Oven ‘a Tims at ia Sn and aul of dhe ended role 45 Aor show 1 Chimieal Drew: € ccoPor st Difpuan: ~RAc wat Dilfuscon: ~kAc' rar) at Connection: (Uat)Aceo Cowen: (UAL) Actx +ax) . ® | wax Discharge dit Runtsu: Quoraxst Qu) so dhe docharge wd dhe ran, Tp ek 2 Lunath. Pov amd Tome. News; Dacuace fey Anees a = macs mm - maso ad, £0 ecttyPaxat =[- Ac tat - (Tat) Rees) 2 eA - [-kAcioerax)at (ak) Acan) + Quast) Dwnduny axaf and Litho AX>0 4i td ket -Uc-@e == Qu Ea I5.@ iku= KU gue uth) o THE (TH $llat+ C)=Tl+ Cet. M(0)= MU, = thc gta C=u~U, po Mit)= 4, + UL (1-41), 6.@ Sr= SG, (1+ £)™ = G (141 YRRE = § (14 Ly" 5 8 oF namo, Section 2.4 Lb) #=6x+5, [4= (asp), y=22045%+C, mo)=0=C, Mx)=225K © Srvayeo, [44 +4fox=0, [A R, y= ehh ca? Hi) =0:Cet ¥ guid C0, po “0 =O. 3 © MN=(y=-me Sag =[€ dy, poial froma “(5+ fo e+ C, = or x 2a] =e" C, Mo)=2-> “In2=C, lg 2 Y= ee) ) y= nm -0<x< b(n), J -e! fsbnn | SHIEERD Balbnz)

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    Section 2.4 8 = feys, gob = blk ETE = Ie, Hh, ENT gio Clas, 20° 4 3 #3) 25-25, m= Lr er oA > +265 Kor=s0 Gnd > +00 an 13-28 < Hp, wa de, ao Xr LhnlS +5 x 5097 fromthe LJ. Som x h) w= ylay [Fy =efd LA bya. 3 K vl Ln fo comemsnce Cm A Thon Salm) ZC + LAC, Inlay) = In C2, Lng = Co, y= efx, Muze=e">C=1, i sd My=ex IE 2.@ dare (FAYCy, 2) - 352 "2 4 e2pl- 110) 20, 4(0)=0}, 410); Gweo M0) = bn (351). 5. ckteA ooo kt 3. dll p(Uew), d= bdr, Inu) = 414A, my=Us = Ue CE. ot . wl yoy=u,=TU+C goto C= Mg so uct)= Us (us THeE*E 5. M'+hm= white pq ane noire eonalombe. 2 = dy NARI EY ty JAK Ce macdun nE0, Aunt), oR Then Siem i i AY = fon Parca NL SEY} N-f= eft ks ) Ppl-nIx PAE : co, age (EC) G6. (b) §'= (6x*1)/(11), (91d = (ex), m= =2205+xX +C. “(0)= 4 apie Dazorosd £0 av) Hy @ex di) =o, m= 12rd +49 | Of thea Jiwvo adatoms chore the + so m(o)=¢. Thus, t= [1+ [FETT TARE 2. k 0 9 9 @ y= i aparedle, = Ang) ro mk. (b) m=wx, ld = 50%) gpa n'= SIN - 10. (b) ‘= 2M, 20-1 foal = 2V-l-NTk 20 8 "ny PT ANE $0), Ao wz = TS W=2)dn _ dy 2 Py aT] i aa wear Cx 2 zhe(v antl) ZZ 2 hE v * har C Tg I 40 = (13H CE Yn andy pmee ZZ SSS Nol, Mm= {3A (AC) ZZ he = E40], [4,2], 144), (46), 1nd de x seme gy il Wi : Er Ln ue Yamada, namady , a0 paon from dhe ODE w= (29-x)/(y-2%), along the dns =2%. Thus, mabiad, red PLE the adudion eto 2x ER Through the swprsndctivt h ponte [4,-2],14,0],14,2) [4.4],

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    Section 2.4 9 In,s], 100], 10,5] [0,10], [4,2], [4.0], R74 //, / Eat Fy-2], [4,4]. For tach sifal po - i FTA SEN is Swe. (y-2%)"= 3x%A, A= (m-220) ~32* J ia Re A= 52,16,-12,-32, 25, 100. For toch 4% ESSENSE Ea hea wnt + and Him - o's 12. Cuma, == 7 — he moments = 2500 4m The Commdind ra il solth (phd): ys Bh wpe XpAR (T= 22+ 9p (2452), 7 7 7; = 2% x= pq (X"2452), ard tam mL : Fer, %=-8..8, 42-1212, Tumpowts = 2510); AL thine ane gape wn dhe Cums urine the comme enema dhe die m=200 We howe Fed wn Shere gapo Jy hand woth da. The Two aovpuplilio yw ont fn = HEL (phen ao —-—) te impoTand . dnvthe sugeens Tama IC, bbiirerm Huse sorgrpilio, thaogh toch tl pot hone sts a somdgue soto’ dufinct M —eD <x <0, rueh ao curweo Thancah [0,10] amd [0-10]. But cmacdan andi plo wn Tad TW: Phung snes a somigie Adio sven qe x<20, Phrgugh, Q thee io mo Adudion (w'= ethane), ond thasugh Rfhene txuls a dhige Adudion own %<%< 00. Sumdanly wm XN. NOTE: The preceding problem, 2.4/10b, or one like it, is recommended for discussion in class, even including the problems encountered with phaseportrait. IL. (c) With x=Uth, =k the speed = (-4)/ (+48 -3) Mecomsa 4 = 1-nr—k 4 AR =0 amd Eis OR and =f. Thon wrk rave qk-3 - . - " d= With ar=2F, v= uw, the Ldn beemsa = dur Lu _ (dw mses - iy co MG = - AHS so [Lr go (da £0 bn [wr (i+ 2wY]) = = Int + (52 C Feooninee A0 * Ln Tur (+2007) = Faro Cie Hn 5 : Putting Jrock. w= N/a, mane U=XH amd N= Ml, aa N 20+ (x-3)y ~% = A N * | whine A 5 ant criland can de * fon x ao a nual aluad func 4 of § co Ascblerluid Jonctiod 4 x. The Adlets 1a onda 73 fe a. dite d, wn Exeneie 10b amd, can be ihuminatid : §usithon, mang sph BpLL

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    Section 2.4 10 0 -1 —- in N= _ -.3 (€3] " = i Let Tray po dor = ead = 142 2 Twa, pri J dw = [dx 40 tury Lhacan-e)=x4C, er, ata) + bn (+n -6)= 3x4 ied the Aba son sified form. N 12. dat =kNF, NPN = kat, NEFs kE+C (p#1), N= [e-prrep]"™ Fon pet, Noto [ipdkE]™ = tf akine p= 18s =» S40 070 ; ni eo BT AAT ppd Fon p>1, Nit) = Th-oowkTH 0 as foie 0 Red wn Tome 4N, svc No= AP goeo f= Ng P. Thue, Nt)» ao £-T, hare T= 1/Lep-0RNE™]. _ BN 13. dN/At = @-bNIN, N©O=N,. Wcth X=at awd N=bNA, L aN sya N SN On Nioy= No = 2 = (ab EEN ou d= 0-OR; oN a No B= AL , boii - dR) =R4R, fis = ce’, Noy=p gio C= : Tho, Niti=- Ce __ Fer | 1-Ce* 5 et __e | Tere Me Lt FET stand for force, bungth c . amd Hime. Boy Tuadm’0 2nd Laur; Mase do moll amdufpenndind : mace = fe = FT x i m FHL x £ c FT/L + F/R F F 8) Vi % Fd . 6% XT To mendumensemaluig T me mud a condrnadim 4 the ponamidina thal hao wonds 4 T, rch ao Vi, Xo/f , Mic, Clk; he choc do mK diigar Lef mo mane Vid, pe. Thed 4a, TX =o. To mendamensconalige x me mud a comdrnation A the paramiing that haa ans A ZL, nase acs Xo, X,/0, F/R, amd 20 en. Lell aio Mon Yo, pony: XE 5 Netimg thd dt = §4T, nang Qteqman Be dew RE) + cL AXE) + RX EE) = Fan 5 %,E0)=%, MIE Fa WERT CIE (x) + & PE Roe Lax

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    Section 2.5 11 ou MW, £x + Cr IZ 1 kot, % =F ain ; %(0)=1 EO Xe [ x* [4] _ _ _ zm me a Fee * (ra) =f) Ho, 0 (5) oL I 3 - ~ i. - ~ ~ oA) premdinc 33,8 salen dhem dhe regal em (hominid) poodle. How Cam ant. nea thal o,@,%,§ ane mandamenaconad 2 a) py moe the fuel Heat AR Loms an dhe final squat (on, amdeed oe he ) mast bane the Dame mmdls, Sei 2X /AX Ao cw he Fr the thin Loma, and wmTial code. REC ee alization so mE magus. Hower the fons mami puindiimcraind Paramdlin iy igo, 1.2, Andon othe chico mode am Mendon tunsonaliction. Section 2.5 x3 Lb) My=o,N,=ov 2E_x*- Foxm= fax = £ + Ay) s U * Fons o+Aty) =o Ao = irs F+C 5) Fo = £4 52 £°C = emstondt amo a) = s Bo. Thun, M(9)=-1 Gato P-1=B s0R=1722, #0 % + Je hes ) Mg=1, N71 Vv E= ed+z ~>Fuyz)=f(e¥+2)oy =e +R - B3E = msinz = +H (2) 20 Al)= - fouiedz= con so Fiyn)= Enz + ez + C= cml. geo eeyzrenz=8, ay. Them, 200)=0 geo €240+6r0=B gues B=, 80 ¥+yz+enz=l. th) My= cory + cRX, Ny =e tony ) ai it AF/ax = frwnty + Mex => Fix u)= fo georodn Khun] Ch 5 y OF/ = AHH + Xp) = Kor +d + ) Ao A= fod Ao Fem) = Xam Ain C=. geo hh oy Them, MR)=3 geo 2mn3+34n2=B, RO Rung +N XT 2 in B+ Bm. 4 Moz 0 My = R, so the squint de eel of Asb. oe 5. (6) Mey, N= chun, My Ny. My = todosel = f= fing x AO T(x)= ex = ero y 5 nde aias re Ft bnady=0. OF/ox = 4 => Frxy)= [4-3= mbnx+ Ney) rg REx Ao he C. Thus, Foxgy=mbnx+ C= emik Fuca adn = 3 2 mon = B/ mx. hax © M=1,N=x, My#N,. . MyNoc = ol = IN gx dene, maze =% Thus, seals the ODE an fdr + dy=0.

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    Section 2.5 12 SFhx=4 = Foum= [ox = bat Roy) 3F/om = 1 = 0+hwy) 0 Awmy=m+C A Thus, Foun) = Dnx+m+ C= cpl. Iritn=8,, HO= tnx 4B. (Ah) Hoe, “tao 2. Mzl-x-z N=1, M A Me-Nx _ 1-0 = _) = gm. 4 2, so Tl) = ef = eX Phas, pede #% 0DENae, E*(-x-2)dx + €* 42=0 Fhx= EX(-x-2) > F(x,2) = [ EX(1-x-1)ax = € (x2) + A(z) 3Fhz= t= £% Ae) so A= C. Tho, Fix, 9)= €% (x+ 2) + C= coat. far EX (+2) =B a1, 3 we mnch, 200= Bex 5 6. IM (pr-q)an+ = dy=0; My=pelP N =peltty ™M Fox = epg)» Fox) = §™ (py-gron +A Flo = ef - {pea +A ] ) (pet . . Tria 4a AIF), no this dignell gue e B which cancelo with dhe Lbs Livm. m dhe Lt, qaeng O=Ap), Rtg) = emt, Thue, Flam) = Je py-ppdn + epi. = ‘con. “gua { SP gy — [Stan dx = C 3d pean [ey Tio = fp tn med abe Te gn [Reg qe EES) IN S e dx = [d(e = ao He 5 a rt th so He Ey ppd ET OT Ty Flory = ei Fon) = elf + Bay Fhx= IPH (4-2) = pS + Bex) gure Bm)= ~ fea +e. 20 Fog = eal. gure mere ef gn + endl. = coal) duck Guo the same , Ind mene. canily. (5) (My=N)/N = (acta —6x-419)/ (sa dy) # dq x alone | Cn WM= : Y/(3xy+2MF)E «Cm ao TO) and Ty) do md wus. Ty T= xing AME(3m + 2 el + Xb (3x4 404 Yay =O nur M news N SAMy=N, de, 3x brig + 2x (pry = 3tar2)% Pre) XH which Cam hi pdiafud oy kg B(b+1)= 3(a+2) amd 2(b+2)= a+ 1), ie, a=lamdb=2. Then un agin ae (a * + 2m") dot + (325 >+ 44>) dy =O Fox = 3Lm>roxm¥— F= f(s +auytyon = N ut Ae Flom = 33x? = 3A + UR + Ny) => Beg=coat. Ao Fx) = Coil. quis the acbutio Kapremt=C. ] 8. The idea, as that FOI + §0)dey 20 Ao “ata, fon amy funilmna $00) amd, 90. Thao, Rog) dot + 10) dy =0 cam de mode wal, an), by dundeng vi £00 amd hy), To dita Td + Fp 44 =0. Thell <a, adm 1/Lico hey].

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    Section 2.5 13 (b) “thao, eX dx — ryt =0. We cam any SEoxz EX ao F= [eax = S&H and Fam = a2 80 on Se, bok io maf in (and sqporedind) fo mend migrate, ; FUE -[*ay=0, [2% + 4= Cot, Hz 1/(Crge™™). . ©) Ardy - dy 20, [ooxdn/minx = [€™ay = conak., dom) + £€=C oy, M0) = =k In [A-2bafinki] (2C2 A, fon crmenionce) 9.(b) (AMO +1)dn +7200, d0=0, My=22e00=N, ro wed. M(2,8) N(,8) Fon = 2mmB +1 => F109) = [eraums+l) an = seme ++ Ace) Hho = TED = eke + Alte) ques AB)= endl.) po Fine)= cma guiothe sdulion Pane += Co (eadd nda $31 56) Om), off deavnid.) © (uy-e?)dx+ X(x-e?)dy =0, M,=2x-e9=N,, so we. Flax = 224) €% Foy) = 1 (2x —e't) w= xxl + Aw) oF/an = x%xed = x=-xe? + pres) Gute Ao = emit, po ry-xel= C. 10. @'=4 (on any mowyano consland) 1 (6) No macsooly. Forex. iY Marg) = € amd Moya) = €9%, Hon My tng) = xe? mrhenvag Mylgxy= me*d 2 xe*d. 12. Fa=C, so patiendon pluto sm Fig) = Fea, b). 13. Dito (MtPhy= (N+), 7 Yao, becsuae & gua PALA 0 0=0v

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    Section 3.2 14 CHAPTER 3 Section 3.2 1. (6) A AX sv Bess LT, po 4% cant be Lith. NO. 2.(b) {a5 x, Antl, x13. (CHn)= (x)= x = + [t= (x2) + (2-1) ie, LHR) —£ (rx) = (x) - + (x1) =O. @) 6(0)+0(x)+0(x*) =O, suns £,0,0 ane ml all gnc Bh) 6(0)-3(20) + 0x) =0 , aching 6,-3,0 ane mel AM gro, 3. (b) Tew Theron 3.2.2: ax ght et WLe™..,e71= [ae a. ae. Buy prdpty Dum Section 104th a &" oe a x [EE TT = Sire ar | Hn & Op Gn . The Ldn Amman to a fre ht | Verdomode domed (ou A Gy Op, Gyn | ExtreselT, Sedum 104) 0 of dhe i a0 are. dune Tham Trl difimmimand, and dumce W (sme dhe 58% fine ane novpno fo all x) io mevieno Bt fellas from Pheorom 3.2.2 Ahat dhe ag20 ane dtindt Thon [€%..,€0%] so LL. Suncdy, ifthe Gj ane mo dst] Fhem dhe at a5 LD. For patpre =z, fon amatimce. Them 45% + 0 —4 dS £0M™ + +06 =0 Lat the effpeant 40-4,0,..,0 met all O. ©) IHX +X? WILL, 1+X¥] = E 12x | =u = 220 po (Mhepumz.z2) LT 0 0 2 ) . © Wpmx, cox, wha] = Es sex a |= = —2pmhx, thick “AX =CpX Aux Ko ood sdurtieally 0 on amy whirl. Hower (Theaum 3.22), LT. [€9) 2 R - _ RN Woewe)= [YJ] = 25 ache do nik idiocy © on omg wind Hence (Phenom 3.2.2), LT. Simca dhe ane wl To for ah ad, A do mipniflan $2 woe Pheocum 3.24: nadhun, 46 & Aeallan multiple o The hen ; Aunce, they ane LT. @ LL Ay Therom. 3.24. 4, 7b) 2x C2 Wem, cn2x] = lees Sonal = -2%0 no (Thm.323) LT.

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    Section 3.3 15 ©) (0 5m (0), we AL gma She Aah found rodent dhe furlind ane andud tLe: £. = |e* xe* ~ Ye ey | & q $e ODE) W & etx” or] Ce | +L .| e* sexe” et | 2x 16 by prep DT (Sudboniod), = €°%(9) #0 20 (Mhm.32.3) LT. Of come, lt dex mud ree DT, 4st cold. punply ig 5.0) Wi = [B31 = Gti-ging)’ = dal +aigl ~ 9% ENT We ; So , _ = 9 Cpofiopco) = Pipa Me Pre Rit R20 = PYM =) and M3 + RAH RYO =p 9 rh =- We 7 : oF 1% #l Ag - $3 pont Phim, (9) an Sedm 2.2 guia Wen = Wig) © 78 oY 6. (a) HY * ane LD thm Gu,0+0, M,00=0 (mT) wth 4,6, mk Ath 0. Yun G) Gd /N Gy BE NMPWRIND. Lit 2,%0, say. Thin At com damit by a; and oda Man == Jann), 20 Ah “o pprumle an a madtiple a: Comedy, pgpee me (oan ih) Com he anprussed ao a mdiple of the © : y= ota. Thm “1U,00 = lly (X) =O thang, md dsth tffeeao ane no (vice the first 40 1); Aumee 4,4, ane LD. (B) Lek U,0)=0, poy. Thin surly OU, (2) + 5M, (x) +OUs(X) + ++ + OUKK) =O wth the crfprecimis 05,0,.,0 amit all 0. Homer the at 5 LD. ©) Ginx) ++ Gp U0) = BU (x) + + Bpllplx) gueo . (mb) 00 + + + (n= BANK) 20. Smee Lappy thn, ae LT, i follows tht G-b,=0,..., Gy-bn=0; ie, Q=byy +, 4n=by- 7. No, && deco met fellow. For wx, 14 amd x ane LT (Thm 3.2.4), 4 and [+21 ane LT, amd x amd +2 ane LT, mid {1,%,1+2%3 40 LD adnce IN -2(x) + 1(1+2X) =O. Co Lo 8. No, because. the Hagum dato md apply sce Bo condFima ane wd wl’. ) Speegcally Aly, p= b/g amd Poo) = 6/% are mk codimimio om nfl cotamnuiy He pout x0. P 1c Section 3.3 1.(b) e*e* amd ° we biliary (sn i saab oidud fyaudettition) and thay ane LT (ome so md a madtiple the ), so C(e-e*™)+ Ce” Aa gemanal ATE - ©) 5 2% io a adudion, hE tt mud Too LT adidims fon a gamnal sdedio. (©) No, we med Hhoue LT sdutima . # Vio. @No (B)Veo Veo

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