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AK-MATH225F10-EX2

# AK-MATH225F10-EX2 - MATH 225 Fall 2010 Exam II ive full...

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Unformatted text preview: MATH 225 Fall 2010 Exam II ive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all In order to rece provided. Where appropriate, please enclose your ﬁnal answers in boxes. reasoning and work' IS 1 (15 points) Find #:113th Lap plicen transform of the following functions. (Use the given table.) :2 L1,,» =3 (a) f(t )— — 3t sin(2t) + t3 He) 32Ht)315{ gas—Lg) (b) W) = (4t—3)U(t-2) :1: (a «>2/7Uc)=%'3 GL5) = 322(1): 7— a (c) h(t) = et cos(3t) * 1 131G Hcs>=§251=w3 5&6, (Oslstﬁ Egg: 2 (15 points) Find the inverse Laplace transftg’ﬁiR of the bfﬂlowing. _ 25—4 l. , (a) F“) _ s(s + 1P Fig} : ‘HZE’F Li (£5) +4,» (Elba) ' iii—L ~ Li L .» 1‘52 #sﬁ»; an»: E' 6L5“): ‘ 5; + 5-H +5407- "‘ “é, )1 8 +1) +Cs 23*! s .1 Ace“ + “5 Hi) = E "< my: .14 (\$2 +2913 ”r @1535) ’1 Ce: 23*! (24+ Ms: + (24+ B+¢>s +4} : 254* FH\$>o Zﬂi-B+»(.=Z ﬁi’ 5 = w :w ¢=z—2A~B 4 ;=(, s_ (b) G<s> = m ﬁllqsntﬁ : 5’3 Ha = L/L/)Z—L/(v)eo CS ‘, 51~L15+W =st—L)s+["~§:t)z+»‘7~['—§sz 7° {5'2}? 3 6—5 '5 l (C) H(s)=252+8 : 92-: 321%) .5 Z H'L5) : % (911%) / S" 25:33 :' , . M M19933“ 22445)?» 1% ode-1 was») ,. _, "Mewhh _ 3. (20 points) Solve the following initial—value problems with Laplace transforms. (a) y' + 33/ = 13 cos(2t), y(0) = 2 4 3-K. 5527’?) +3327? 2' 13 gngZ/ZQE if __:>_._~ :_,7.5 543%; #1) ’75 5"! ’-l :21: (04> 1* 4,, L= 2 , A [524q>+(851¢)[s+5)135 L was 2435 TC +5655 s has) «a +3 ms) ~ 331 45 T 5 5 5L } Laws 4 (séfckﬂh Was) (234 113:) 3’65) =1 Z + 2;; ’35 Z. 3 A+5=o 315+¢>13 H/Hscas : + ._.___.._...._.l S 2- v Liﬂ: :Vl'ﬂ Vb) 513 (.svaMs‘M) B 3:: 8‘ C 5 — 2....7/3 - VLS)’5+5 —.§_ +35+H 5’3 3” C’LT 5+5 sis'r‘l fr' ”T 5» '3 +952 «L40 <>y”+%—m= w—n 3271’: +2327§ 3 3277: si 5&5 a); aw) ‘ #lla) FF’ZD,%=’; e Ws>— —€>(c) arzfs‘ﬂa) 03 ~3Ws) :63 Le +25v3)‘fés) = ) +6.5 YLs) : 63:75:; +5.: am) \ — A» R E [§—t5)(5»/) '53“ 1"“ f A’Lé ’0 +81s+3)>l ELL H84 ’ﬁ' , \$93: 444:)! _; ~21L \ I . 1L + ’5 _L L , l VLS); Vt:— s+5 ﬁgs—I \C L) Srs +777 54) fl; 3/“: ,3 it 3/0; :H lY/Rll J2“), “Tla +11L6 ﬂ” WLDELC Lag, J/ 4. (10 points) Use Laplace transforms to solve the following integral equation. t f(t) = 2 -—/ f(t — T)’Td7‘ 0 game: 1 \$223 — 33 tile m); #2 #19). ms) : .52.: - 321; face: \$937M 5L 5: S __ 2. _. é a _ PCS)” s 5-H) ‘2’ sim>/ TCHL)~Z<°575 7* 4., A= ’ 5. (12 points) Given the system of differential equations 2 '10 d3? = 2m— 10y A T” > 1 ~ 3 dy 2:4 ~10 _ = _ 5 ‘ ~ dt m y Ami 'l I “5%, Find the general solution of the system. _-_ ‘ 2w ..0 - ‘ = ~ , 1:“: 33* A—r =0 ., O o , Jo£( ‘ "5~A.> o A. [,4 A; iv, o )1; K «Jug (2-43 tam) macro :— :golzc;>%f) (? 1%: EV]- JOVa: v =\V’ L z 5 l3 6. (18 points) Given the system of differential equations ; 2’ __\ B E_ 1 -13 ' I'IK 43 — X dt (2 —1> A‘Ktlz 2. 4*» (a) Find the general solution ‘.. g "“ "3 = Af'sl: (Ah/111W: :0 QC? 2, ~|m O x (V51 -13 .>[V,B:(o> LwOCva) -[-13)(2>ﬁ€> 2. ~l—5L v; o AZ-VZSVD U‘S‘Livg‘43VL—TO 7.1" 5 _S‘ A, 2‘ V1: IIBL v, A::SL VI ’3 <L%i>/ V/:[3/ VI wé‘sL ‘l: s t '3 ‘ \ Is 6/“ \7, = C L (i’5L> (Cosigtﬁ— 510615)) (1‘52) ». , 13 chLS-t) +13Tsm6t) CD451) , g: CcslSt) th?o(5#t) Jr 5577 (so) ,3 c¢s(5t) . ( 43 521’) Si) caséSt) Jr'Ssi‘bKSi) + L ~§¢¢5[;O +576ZSI) WNW Mmm-ww "’ v I 5 C05 [51> ( 13 57') (St) I X (i) v C‘ < coSZSU‘HSSM (51)) + (2' ”9C53(5%,)'+Si015‘t)\ / mm_mm..m V W’w'x'vrmmeVw-WmnvxWWnM~4Im~wrmWWW.~ m, MM w W (b) Find the solution that satisﬁes the initial- condition X(0)= (13 4). Report your solution as one real vector __9 I3: VKO):C1< >+<az :j (’3) iBCI”/B MSCL: C‘:‘ QCL“’H1C-i :i lacaszSt) + :34ng chészso +(Dsi:(5i) 7. (10 points) The general solution of the linear system X' = < _21 :15 > X is given below. we>e4t+c1<i>te4t+<1>e41 (a) Find the solution that satisﬁes the initial—condition X0 =(1 —-1). Report your solution as one real 7;; LIZ > +cz U >11}; l (b) Classify the critical (equilibrium) point 0 0 of the system. k: ”LL: ”>0 I 96201“ < unset/c nook (c) valJtu-sns tl—vd: Aft, not 69ml limp 5b(o'f:1\o 017\$ Lékmc «H tAkb' h 5'); i approach (0 O) as t —+ 00. FF @ove away from (0,0) as t —> 00. iii. are periodic. (d) Circle the correct phase portrait for the given system. ...
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AK-MATH225F10-EX2 - MATH 225 Fall 2010 Exam II ive full...

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