219F - M E T U Department of Mathematics Math 219...

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Unformatted text preview: M E T U Department of Mathematics Math 219 Differential Equations FINAL 12.01.2012 Last Name 1 Sigflfiture 1 Section Name - Student No: Duration i 120 minutes 4 QUESTIONS ON 6 PAGES TOTAL 120 POINTS —i SHOW YOUR WORK (5+15 pts) 1. (a) Verify that yfit) : t is a soiution of the equation tgy”—t(t+2)y’+(t+2) =0, we. 3.24:- 3*; 3‘35 '3"; a 43. o we Le+1). 4+ Ltrkafiwrxa Tfi 0\ fiOLUA—‘Oh ‘ $0 (b) Using the method of Reduction of Order, find the general soiution of the equation tgy" —t(t+2)y’+(t+2)y M2153, 25> 0. Identify the complementary solution and the particular solution of the equation. ? at“ W“ ‘3”;2o’+ev’ 3 (lVi‘V'EVH) wt" C£+Z> (wed—AH) + L£+z>tvg2t EVN+L1432 ezcefl) V/flg—es ) {EVI/Wztgv/ml'tg/ Vij :2“ Léfi wavy finiém UL fiv” cm .i. 5H: mlfiLLd-zzl / u, ill—Pvt) DArULw 2 (14+5+7+14 pats) 2. Let A: [ 3 mi] . (21) Find a fundamental matrix for the system x’ w AX. Ae€CAwfii>$ ‘ (L NW any"? I”, I W: mezbfimfi’Z) *3) “z; '7\ w—l :_.Oj7\—w~“+l 3 rgwfi L0 \ wag, g wlxl W31 '1 % éfgzw ’ I [3 -w$ m) % ‘ X I“— e’ 1 cm 1 “I 1-,; 7\::-—l 3 “l 331"§>;.3G} E); Hg 3 w\ {2 wt \ J% (3 , A ék .Jt Funclcxmenlal m0x+flx Wbtj: at 3C2, (13) Which of the following phase portraits does the sygtem in part (a) have? :5 A “:33...” M><O \\.'f-.\'\\.\\ §\\\K\\i §\K\\\\\V §\V\\\\\R\V ' (L WR3EfyljC;B;Xmflawm)[:;%;ix;liflgl X5“ (32 3 3/2 .._ - i ) 1:, E ] Lm]"i* 3 ) ][°]:W-£w" “M C2, “’ L ”\ 1 ‘ t W , 3 “b “L *‘ #4: w , . m X; 3/ (3:17 33 WV; L éefiw ée (d) Use the method of Variation of Parameters to find the genera} solution of x’ m Alibi—gt)= Where g(t) r: ( i: > 6"”, by using the fundamental matrix you found in paEt (a). U) Xvwwim w:— w‘cdi‘flww Va] . U4) . W U55 {4 2:. A—LPLk) LVL‘b)£\/t 1:: AX?—\'3(:{:B 1%.] ' "1(3Lt) ’ a 3-35 :1 at? U/ m W F“ Wflfl[vd]w3fifl’ [W]332;€E a: [Ea _ O ‘01 r z: m 3£38 “39’ ]~;Ll I v] ai , ‘5 (1 WI + 3 Mt Mr 0 [ ' 6” t] ( KP;L1JU:)[JC]W 3kg 0) 6 xfi‘fl _¥ KP K;C\ >4 “)t "2/ “:6, w¢[:\]+ -b@Wt wclafiLE'lJsze-w 3 SEEK. 4 (20 pts) 3. Use Laplace transform to solve the initial value problem 9” +19 = Mt) + 5(1' - 3), W) W 0, KJI(0)=1- ” __ e? . Jr Vang“ } i§é+3r§~?+e’ ' L6" 3 N. x. W25 "WES Sign/:1: Q’s «ft-EL -25 -335 (Si—143%: ‘3’;— + «2 +| 5 “.25 W35 3 \ GEL a JV 2 + / 5&5th 32“+\ 32”“ 2 Ag rewa— ‘..2. 5 : flaw-Jr '32“+\ # EBCE’Z'“) $(§~H m A+%=_0; are) Awb was 6.. \/- e’ w" {32* + “2*\ Mk 2‘44 “" 5 5 +1 (3 + 5 CO 300*?) -+ v13 tic) am C£~3>~+$¥n41 a ,— uluc) .__ Ulla 0'! (20+10+10 pts) 4. Consider the boundary value problem (BVP) agate, t) m 611, 8332 m (2t+1)még , u(0, t) = u(1,t) = 0. (a) Apply the method of Separation of Variables to obiain the general soiuiaion of the given BVP (IncEude a1} the derivations in all possible cases). .m/ ’1 7"“ w may): XLxFVLk) I UL“ :- >< \} We W X \ ” 3T” , i X (2_«t—:~+~\ figfl r; _ W “- >4 T: c1t+\)><T J X W. T ><”+r>\><-.-:O, C”%+‘>’TJ*“T;O oXW—O “03:0 {'3 is); X(O>TL '” 3 ML‘ ~ 5-3 XLOBO ucwc): XLUTL“: W J ——~ 0 XLO):O XCU—m e 50:10 ><I\W+"7\X:~O} If} Xfla‘xfi’b f X835 \ *0 , X 5W0: A M O W Ca 5: 6L M 6 (b) Using part (a), find the solution of the given BVP which satisfies u(:c, O) = sin (37m) + 751:1 (57m) . m n"~ 7 an S1 MLV‘WX) m3: Chm?) 1£®P “#315* 53h€3fi>k3 %‘1‘5VHL5WX) __._ u (xx/o) :1 ) __z WSW—272, ""251113 u (wt) :2 (ch +l> ifncg'wy) 4r r? mama) sfngmj (c) Find the solution of the problem 82 8 2%” m (2t+1)5—E, v(0,t) = 0, v(1,t)= 1 which depends on :0 only. \fCX)bb> ‘3" '(LCK) 1 - .._ \[xx :27?\CX3) kao “3.53 1C frL x) -_—, Vxx :- £2364“)th l J? (>43 7:134 «C Lx5m- Cx KW+LZ :: Q X ...
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219F - M E T U Department of Mathematics Math 219...

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