MATH 202 Exam 2 with Solution, Spring 2006

Math 202 Que Seo om QH0S- oF Sec ons \- 4 (Prfesser Komal Chori - Maka: x) 1. (16 points; please try to solve on this page and the nexi one) Find the general solution of the differential equation 1 eo Hint: part of this problem needs variation of parameters. You are allowed to write down the system of equations involving wu; and ug directly, without deriving them. For the integrations, J suggest, using the substitution w = e* and some partial fractions. y” + by’ tayo dt sinde + S Yo: cvs any 4s m4 Suet wely 1,74 - Ae™ Hye Me= + (os Xmre eo e Deedee Sey whee L Gap f= x3 Lye: 2-2 \ Greg) = er | Up |: LOIxj= Sethe L (An @]< A (64.9 = (ShA4BD 4 HAs Rata xe fh. LO V= 4 +64) LELw\s 4 a \nee | L 3 x Va Mac ( “* - B wh . % — li ia hker Noter 4 Reyes yxe Sk, ‘ Sey Lfet}- -iont 5 Lf-s ests | ee \ Yt vos Dm ( LCs Yo\ = \vew x) nn ceed 4 ‘ : wh = 4, rd. qe ) wean PH Tee J Pe Se re ye fort fay uy ge (DE ig trod) x | Hy tg 1D) ome oy aerating (eo eu Fre “wy = ~ fen Yeo | . x, “yg Le -ey, -He “o) < a @ “36 “rg GO => 4, r = $Q =3 o/s ab a ex a dua e” | 7 es & F % iw 2. (16 points; please try to solve on this page and the next one) Using a series centered at x = 0, find the general solution of y" ~ 3ay'—y =0 Meo ke on ordin + so Mek c = “ ' oe Ret w_ °2 aD e af ax < fn = EB efenagx eS (pee Os ; nro , 4 Aro ax | 1 hzo 2m S 1 & 4 = a- DE Lecemes. 2 n(a-D eo, x + 26-Dn CO, + en) c,x” =o ‘ oe eed zo ne combine = n ad o 2 nln t) tn _ j (Bn41 en x2 oO mo nto [eek cue (*) See lela, \~ ten «< dees koh = kCe-1y ce xe ~ 2. BbDe 7 rt 2 Oo Ne feve = h Bx -5) Seve sine POUT eA CE kro = Ocgre fone me k22 => Me-da,~Gk-s)o,, =O lez) =p O20 sy cee a Be < Cer | CRderm reunmeree) L a oem aN names GS 2-5 Ce C. AT Cu oo~ Oe. (8-1) Ce 0 Oy Ceegy en: \ ros a Cyc he a ob 2 DA T D cy a, Co, Ce ae b, Cx oS e 3 oe a2 eu a Ss Cy Coes ims oa" AD 2 is ‘S$ = ¢, Cees {k: 5) Ce 7) 4 al Creag” & Y.lo i $ | Sa RE, SEP A, cae Holt _ tole. (69-2) TT By-2z) | . : ae tare ! Capa = (anys %y = jet Ze, Pe ’ (WI) i a . on }- Gt cee cyte egyie = = ali+ abe Wah DIB em = ai 7 " OK + cau +e gx? + 9x?» oe ca) ‘ + re. ten mS TH ta, HB Miho VS ato 169 10 te AG 4H Oy] t HKG Ke DE ey a Plot 1b 9 ne a! ayn" 8, a * xt) x, ad Ile >) an g (é 2) o., | bot J T,]- 5) 28 Hy, yD ater \" ob ental eee So eee x + qe JEL \ Qzo CH | fo CD L = nates a , lie obit the bet Ricoh He Ded gek ae Rema v on & ‘ $ G A L Set show ee, S (eerteedo, x — kaqx to =) aes x a ate kep Be recurrence Ca = (eaten ryt {, k20 ted 3 Ch ance Ws leads tae coun Ahn) 3. (16 points; please try to solve on this page and the next one) Using a series centered at 2 = 0, find the genera] solution of Qa? y” - (a? — Bx)y! + 3y = 0. were xe ee Pe Seer 2 Se ea 3 xXco is epvlen singer b, Dy ,* ’ R ae Xx x ° p x oe den Shin dew nar 2 natot ue (owe ry J YE EGO Denn Neb We Use. 4° s nee LE aden & i. an aan En hee =o t | . a aot ~ I _ Subrchvee in 22¢ i “bxy +3y <d of nett OA nee aw ec, > aimee Dee 4 Bo Cneed a 4 Salmo)ar Stax" Reo are Are heb co Li _ — Tad be , ere Scr teh & £ q: serps oe. ae > CO OEO nx 20 iF Ucate we haw o Coe ° 3Yrrr-) Ge + : Qde (aoe AN h~o ayiek htt) Quiy-l-2us B ned ok ne Fo) 7 > Qe -Su43 { Ce en) | seser ke! =(0- 30-1) ray em kar kar a x =O a. (2to-3Leer) qe E Chew Dy, keo : ket PE Corner e _ eee mae — Wee -d : — - a _ Set boo 5 (2r-3te Me . kel oy C2 - Sere, s Cer" De, ~ ° a iT to 4 \ rsh ow ce! | a es BH ete nla et) whe Hay bat “pd kate ite 20 rede fet OA Cre ee woe Ber hy Yrle gL NT om NGL, fe We wil afk Q shes nities) So are) fo Rive am coral HL. pawns Ee ont oN cancal ik Maher tasheed : Peweves, Lu : nwa oie. o ke As osWvh) re % | emurren me Sy (206 %)-3) (4) ge (KER -D Ca Be \ \ be Fales kal te cm AEG + CREO be Ata ~ adel o cnc wy here y woes Bek Mga th gg |e bay ‘ Be -| dt aa Hae Pre = a a Cc, he —s__, yn. m— Ge," ch > Ged? Sr Gay Ceay 3 Seng 3 3, 1A yy i <) 1 \ Robe. we om, rwahe 4,= % a 5¢ G1 20 ‘ 3 — continued. he selolan | Tel recorrence Sens (end syCeu Ne, Ckat-0) c <0 Rh ke 2 | & Lede Key 20 Rekzi seber oth on be emceled SS \Cy ~— wer _ 2e-\ | . t c t 4 % Be ° Cc — ie Ck) ' fhe “ ae) _:- Ce Ch ‘4 (k= 3) ‘ees \ “lo ~ te . 7! ne ci Te 3 pase, Ct ie?’ - aoe c, 13-5 eT sale, Mae cov ac, sept a pete _ ie Co (arse check Yast “Foy ae Gaol “ef, Git roan 1 ZaHl l eae @ am Km Ue . -t a Bee em oa] 135. - (ey No Y CD Qri Se ao % 2. : . 4 - Qa! Qro . _ -SowwTen 4 = Ay, + By ee a 3 do \ | - LARC Coe Rey | - i = a x +R tam —- %* : | Q=o 2h Qeo 2R)' Le — J 4. (8 points) Using any method that you like, find the genera] solution of ba*y” + 82y'! +y = 0. Tha see Covey Ber equchen, a . Sam ye x, oe atk oie ® loki A thy i) ‘ a Ser -1) Xx Ne rex 2 © te es onlin ate. > FC -Uviee & re s1Eee L Vann salobens \ = omen ers 5 tek - we here conplee Sohn : os 1( 3, ig L Es) na ag S 4 gu x (e > x FC es (kid + ‘4 (% nx) Ye * * Ss eek = «8 (alex) = (9) [ ws wall make a shhh xee= — Cdetails WPL a p>) ‘ . 4g 5-2 = p43 ogre Cs emt uxt “ Ink now ante elt) ‘755 wth rocks Me Mat 142. oe ouxiliny nh 5 = ashe alto + Bea (%t) Nn (Kd Red mm (%lax) 7 5. (8 points) a) Use the Wronskian determinant to show that the functions ¥, = 2, yz = cosx, and ys = sins are linearly independent. ; 4s 13 5 : WG oy — pede (TY a ; ; . let ( * ete Smee > 13 Lh ashe Cex " uM ” rn 10> Os Sia oe ten © > pond ale Ick column Cong othe Clece = x ke fo RT™ Cox fULX Ga x wares jor well, ba Mus ig —\. ‘ \ Lo ( easier ) - ree . deg - dlos'b are) = KX. [+5 x ~(-o8 "| - [sp [a x e Ut -Slaw , roMK Coby certs) =x, ; WC ar a See OG, ty ot dekely wee we Gabel oy Onl ya om beer eee Oo cad oder bengentee nae BE ole | yourl ale ig zhx b) WITHOUT using the Wronskian determinant, show directly from the definition that the *he-® functions y) = 1, yo = e*, and yg = e7* are linearly independent. Coie ere we amie hws ‘her Tha oe ® A+ &e% + Ce~* = 0 f, all x hos One or Ve Arwsl glubn owe ABC are al\ zero, Vor metal, ee OD I Lekvahee $9 ; ae 4 dhe, ‘U) A+ Ge%Q Ce" = ee ne @ RE- Ce% Oa go six } These esta => = = / we kee *.ce%<e woo a>) be € mil Ey vad est) ae BYMd vk nel & => h=@-Cee Con. vee Ker), G fay siete A ® mwa be the diye! On Az heCro BQ 1,e%, Om ae Wreerdy rhage, for de —— wt xe! Dd mative a eve Graton ela Seppe A nee, BL, A *+BrC =o @ (49881 coy - Aa Rr =o © 9% A x Re +Gr0 (>) (fp caee KEL) a) - *C(k 4) =0 (Hw “oe) oe, a +Cero@ Chew xe-1) (1) +e) a @ -e/ ey Fe \) +B+C 2 § oe oS Reo =F t- fre “Eom Car beens ye kso 2S ote “exc re Aeinceo we rete Cab Tah ES 6. (8 points) Consider but DO NOT SOLVE the differential equation w?(a — 4)? (2? — da + 5)y"” + (2 ~1)(e@— Ay’ +y =O. Repeat: DO NOT SOLVE THIS EQUATION! 8) Identify the singular points (both real and complex) and indicate which singular points are regular singular points. pe SD) 2 Oe To onake Hee the no of 7 Cl HAY CEA +8) x(x 4) 63 Hy +5) e-4¥+ 5 Ae Dal, by | Shy peels IE ae Mle, R . Bat OOHars) 5 ee Can feck Row oS 2 &-OCx- PD * uM, f= 241 Be 2-1 O Me soul fee one Muse lene oth Per Q ha ao sda of he dare nabers of the one, The Smt te nie Super gees one. Aha supe Penta Keg gech the the dons ai netye me) P conte as ob musk the Lick (x a) be Hee densi noter & Q Contes GL math the aur. facta Geay*, Cte eth aqomrt. So fe rellen Soper pot oe welt) neeei, node | b) WITHOUT SOLVING THE DIFFERENTIAL EQUATION, find the minimum guaranteed The ony Weeve rere see PR bee x pee Ke the denon rah, radius of convergence of a series solution of the form y = S p(x — 3)". Justify your reasoning. n=0 [Lok at- the conter x= Qo the Fogle powks ©Ae x20 xed yee) ” The, Complex place: cal @® 7 Me & dences Le 2 earch, sinter pon One. Orcled Ree Ley, <2 rw By Ate Theeren, we ane evirmlee$! a alin Guanteed tedis ot Conve ale. _ (© Cur sewhun Sc, lx on 1s Guaranteed 4+. Cenvere Lf. |x-2\ <i » 9