MATH 202 Final Student Copy, Spring 2017

Problem 1 (20 pts) Use the divergence theorem to = xi+ yj 4 zk across the boundary of th below by the cone = a : find the outward flux of the field e region D which is bounded from above by the sphere p =1 and from surf ny \ Sa clo sed SOT ¢ Qe, Zz av €) ~ SS sikes 3 i eV) = af . Weer Cay, Anvergen Vu of & (Sap ool ut Problem 2 (18 pts) Solve the following IVP. dy _{e+ye”’*} /x dx xe” y(l)=0 Please make sure to write your solution in explicit form, and to indicate the interval where your solution is defined, ais tag) (ce Hee) “Xe e0/x : = lak ea K ax jz ux (° wih v+u4 Problem 3° Problem 4 (14 pts each) Find the general solution for each of the following ODE’s. £93,044 -0 Caan 5taX Coe Ce wr Xs, COV Sponskag ©Sy~ ahion’ Ae $ 1,6 | . d m —osm _kw mM 3 By ~~, Aw, > r - at n og M = | 07 = Wes Y zt : ss SS ~ gant fe . mM = ~ccee a ae in om TE ——— Ju 2) LU i 5 Lg wm tam *Um-b6m: g P Lakr # m( im 4am tlm 3) w) a C yw. 0 ff ; (od - Sei 0 ~ ~—2-2 A os me \ —I\ cele ) =f m ——7fu/ Jae G [ caces 27 tcssinnt] ¥ Maa TeICY r) BD le J =a F \F [ Los < des CO OCR ai a ie. aie NO ee bs CF an tt \ a AG \ ae Se Problem § (1g Pts) Find two linearly independent Series soluti ” tions of the ODE Y —2xy'-2y=9 about the Ordinary point x = 0, (It is enough to find the first four nonzero terms of each Series.) LY a) ia) A t. P pos S cnx Vie Fe ff paan jro ROK CR 58 na Converx S$ On MO p = San N=0 a Ae wie Ss NCA) Cn% ~ Nrzo ia) \ ae oe) S NN YX =U ‘ Nn-0 Aa n a 4) >) 2 cae Caen ‘ U Nco \\2 nN 7 a ro) a — a CaX n 1 \ Cax 1 a z x Sf ns Oe & A ns = Yi—o Tir aaG Ag ) "\ A-v S\N +1) Capa y: 4 Ala hone EE ) Za We. A 9 Zz n=o \ ‘S ele ees Nv > p22 Ki n& eT -2(K-t)uc| me OCotOcy + FF [ K)K-") OK ee Kev i KC Co myth a 2(KAN) Gee B nae a rar ia C\aie= KUL) K seeoveO™™” ng yiserenr 2011 an -4eS ‘i \ se a AN kK Ciro —C6 Coie Migs SC a)aa ot 6 = ke (ke So Yon Co + x o4, (11 1 MT ei! Sf eee Go el mh Co (was € ) collie 8 ff tom / =: (ae) | , / Ma ea 4 Ci X 4 2 2. Cee | ( 3G 3.5 Sy Oils we) 20-1 ay a Ct a ae 12? 10 -a Wh! ‘ — Vy a) aa an Problem 6 (30 pts) Use the metho 3 poet robenius to find ONE Series solutions of the ODI Cea) y yy y=0 Singular point x = 0. 1€ second solution. (You m Wish without the proof.) Then use reduction of order to find ay use the formula for reduction of order if You Wu \ ,sas oa lok Ba NAAN Cog AN) Oye 8 Cor = > Cn? ¥ ie = \y-a <I \e ce / naft_| JAE) ] - & O40 ca x Jejoi ll. a Nn+f-L ok !4a sf 4 = *S (ar) (mare) Ca X_ ” Nuo a [ Nem \ A diy wet ayy) +7 re) (x+y) 1't_ eg Vv 91 lo o r oS . ae va ner Ss (nas) tr ) ( co Shaner) C nx a (net) (n a, ¢ CASE A i pe noc nae \ ) wan = ym. Z Up r)inF ~~ = 96 os erin (n+/) (nef-t) ayes.) Ca S| as eset) ag \ J No : \=o +f—| — rat 0 a S$ ra =O Ze, 7a ( eV fal )) S (nae)(ner En ne ie BR. Welsh n= kK oss ap om Kar yan, = : 4 KAY 4 +r) CK . = Zoe WT imal * K-o in a tr Lire eS oD fj f, - Kari (KO) Cpe 1% S (url af us : Keo Ke So) r(t-)~4, rol,or FO 11