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finalOneSoln - VERSION 01 1 Solve the following equation ut...

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Unformatted text preview: VERSION 01 1. Solve the following equation: ut =2 um, U<ax<1; u(0,t) = 0, u(1,t) = O; u(a:,0) = m- 3sin(61r:r;). l (a) 8—415 8111(432) - 33-6: Sin(533) Oblfl'l'); 0‘14.) (Sign #fiX‘l GJ‘ng‘l/l 67% (b) 6—1“ sin(4a:) — 36”“It sin(6:v) g M (c) e"1615 sin(41rm) — 38‘35tsin(61m:) C] ”\Sll’I/‘M‘X'l‘ Clq') She-[TX / (:1) 6—16” t sin(41r:n) — 36‘36’r2‘ sin(6m: Q 1 (e) e‘4fl2‘sin(47ra:) — 36—6’r2‘sin(67rm) : 5“ le Q (4% $9th *"'-% E‘fi CLUBB‘IA Elk 0 0. (16+) : ~—I {m go) = 0< 33 < 7? “rt um: : 46 I u(0,t)=0, u(7r,t)= ; at“): Q Elma I 2. Solve the following equation: (B) i: E2; [1 _ cos(mr)] e~n21rt sin(n7r:c) CM [0‘25 g0 igfifl “X Q4 ,— {TL/a {wt-F): 2 CML+)§W\ nx ‘ 0 d P‘ '2: 53— i— G» Dr) 9ka 2 1*)?" CW”) COM/“X 11 3: col/EX] W q whal— V)” Cum: -\ Cull”) % GENT—Cpl?) e “,3. 1:- Gang ' ”Kl-*3:— 3. Find the steady state for u: = um + 5Sin(77r:c), 0 < m < 1; u(0, t) = 0, u(1, t) = 0; u(m, 0) 2 93(1 — as). 7 f S‘m 4AM: ~fgtfi'fi} ' fl/xhflfio 7141090 ‘ a“ 44.6") 2 ”L Sx‘iqlnx (e) the zero function fir“— 4. Find the steady state for (a) the zero function i (b) the constant function g r M [fl 9?} % ® (FM Hake ' c the constant function a 5. Find the steady state for (a) the zero function ( b) the constant function 7.5 I 3 £4 a: 10- ——— (c) 593+ 6 12 (dl10—5$+x(1—m) \ 6. Solve the following equation: ‘3 Find the steady state of the solution. (791% gqnva—fisk :. ‘rLl‘ll-Cffi Ca 99931’ P‘ vb“- " ‘W/ ‘1) __Sm(m) — cs) gmfix (C) 71-2 sin(1r:c) W CG‘) 1': __ oCkl') _L‘_¢Er-q2 C El») (cl) —-7rs1n(7m:) & .\_ C + “.0 (e) There is no steady state as the elastic string will keep oscill atI ng 8. Find the Fourier series for the following function: YZ‘ ‘E'Y‘f- fl L1) 10 “FT , <$<§ ? _ .f(=r)={0,%<x<21T 1: if l {“4“} 2 °° 1 1 _ _ ' _. 1 ._ ' + 7r E [n sm(n7r) cos m: + n ( cos(n7r)) em 1151:] 00 1 1 + Z: [I]; sin(mr) cos mm: + h— (1 — cos(nrr)) sin mm] 1 °° 1 1 (e) + ; Z [nsinig-r-cosnmc +— n (1 — cos T?) sinmrm] 9. Solve the following equation: 3;" + Zy’ + 23; = 6(t —- 1),7 y(0) = 0, y’(0) = 0. a. e“1 in — u - (‘5 l ) + S (t 1)) (t 1) 232 a Q, (b) 6*‘1 sin(t — 1)u(t — 1) S + QM“; " (d) e"{t"1lcos(t — 1)u(t »— 1) 7’:— ——l—-—-—- g; (e) (8—0—1) + cos(t - 1)) u(t — 1) $‘Hfi ‘1’ .. 10. Solve the following equation: 8 T W , F. '— St :9 + 83; = 1+ 6 6‘- (J (a) + 38““ — 28—8t \ [L (b) 8 — 26—“ + 68—8: Sacgi> : ,1. . é (o) 4 — e~2t + 38—“ \ \f 3 6+6 (d) %+%86t_§€8t L (e) 4 _ 82: +36% \ r- l + ”L— " 5 (9:33) \2} 29m) 11. Solve the following equation: ) b) Int-”z‘l': (c) film—2+1: \ (CUE 1 3— ‘3:— EEJ IfflJVEDJ 12. Solve t e o ow1ng equation: 21” + 4y’ + 4y = 0, y(—1)= 0, y'(-1) = 3- : 7F?” {if is + C1 (d) 3€h2z cos 253 + (3’23 Sin 29: — 3 (6) 362m cos 233 + 32‘” sin 2m — 3 13. Consider the following system with a parameter or: 8» 431% 3 a X ’ = X _. "- ( 8 _5 l \ Mjlhvr) ix «D Which of the following statement is true? ‘7. a if I — J # (a) As a increases and passes through —2, )\“’;‘)\ ( + > the system changes from a saddle to a sink. + (b) As 0: increases and passes through —2, AK -—), '5; 4+ 60 gm the system changes from a spiral in to a spiral out. (c) As 01 increases and passes through —2, the system changes from a spiral in to a source. ((1) As or increases and passes through— 8—5 , the system changes from a sink to a source point. (e ) As 0: increases and passes through— 1—85 , the system changes from a sink to a saddle point. 15. Find the Laplace transform of the function f (t) which equals (t - 1 2 for_0 g t g 1 and 0 otherwise. ’7. 1—e-s 2 1 19W“ (4") @99’ M930] 2321-31) «Le—hw— item/mg“) -; five) out) _— (:91) We) of a a __£~_."-$ ’5? 3"§+%>~— 839. 16. Find the inverse Laplace transform of the following function: gm ) 3 + 1 : _\_ ""r s2 + 43 + 13 33 5l ‘5 (a) 63: oos(2t) - 383tsin(2t) (b) éeBtCOSQt) + e3‘sin(2t) 5:; _—-——-S-‘iji——T (c) e—2t eos(3t) + {3'2t sin(3t) @+ Q) + 8 (d) 64:00:; 315 + 36—2tsin(3t) ' (e) 6—2t oos(3t) —— le‘”sin(3t) ' I 8+9 .— \ : (Swan; 31' 11 17. Find the inverse Laplace transform of the following function: Q .1) swim 3" é“ gj 9H "l— @19th @)b+0— t)e% %§P (b):+ (1— t)e_t l" .A‘SQI)+BC§+O+CE 0+D§V ©@=* 5“ d) 2+t+ 2+t S3"\ D :D: (e)2+t+m* 18. The convolution of t and 6“ equals: e ( . Affllo (s) (b) —1 + t3"f (c)t+te‘t Hlfll) ;&/ ((1)0g 1+te‘t \ a i 3\ l ()8 W gigg- ”is" *“gfP 51 +80" Align mm t % z "931' “QTY (16%.? #5 S‘IQ' C344) trill; 2\\,L _ i Q l S1 f Sir) AMEN} BQT-OT Q L 3/ §7t=% B”! " win gas-1:; CW 99* PWCZWQ AW] 12 19. Find the Laplace transform of the following function: 20. Find the Laplace transform of f which consists of an infinite sequence of positive delta functions at t = 1, 2, 3, . . . and negative delta functions at t = 2,4,6, . . .: f0) 21. Solve the following differential equation: 15 23. Consider the following improved Euler scheme for solving: y’ = f(t, y): given 3;”, the next value yn+1 is obtained by the following procedure: gn+1 = yn + fun: ynxtn+1 — tn) ri-ai'rL-H = 9n + %(f(tm yn) + f(tn+1:%/1£tn+l -' tn) H'H If you use the above scheme to solve y’ = y with y(0) a 1 and uniform time steps: tn“ — tn = h, then What is the exact value of y1? 16 of the following system: 24. Find the general solution 17 25. Consider the following matrix: 1 0 O 0 A: 1 1 O 0 0 1 2 0 0 0 0 2 Which of the following statement is true? (a) 2 is an eigenvalue with three linearly independent eigenveotors and 1 is an eigenvalue with only one eigenvector. (b) 2 is an eigenvalue with only one eigenvector and 1 is an eigenvalue with three linearly independent eigenvectors. (c) 2 is an eigenvalue with only one eigenvector and 1 is an ei envalue with two linearl ii-deendent ei_envectors. (cl) 2 is an eigenvalue with two linearly independent eigenvectors and 1 is an eigenvalue with only one eigenvector. (e) 2 is an eigenvalue with two linearly independent eigenvectors and 1 is an eigenvalue with two linearly independent eigenveotors. hflfiflfld Hinze lTEQA—JL s] does so fii‘fle‘u loop O\QO onob 0000 00c) ...
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