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Unformatted text preview: MA 303: Differential and Partial Differential
Equations for Engineering and Sciences
Fall 2011, Test Two (Instructor: Aaron N. K. Yip) o This test booklet has TEN questions — EIGHT MULTIPLE CHOICE and TWO WRITTEN questions e totaling 100 points for the whole test. You have 60 min—
utes to do this test. Plan your time well. Read the questions carefully. You do not need to attempt the questions in sequence. o This test is closed book and closed note. No calculator nor any other elec tronic device is allowed. a For the multiple choice questions, no partial credit will be given. For the written questions, in order to get full credits, you need to give correct and
simpliﬁed answers and explain in a comprehensible way how you arrive at them. a You can use both sides of the papers to write your answers. But please indicate so if you do. Name: (Write Clearly) Question Score
#1 — 8.(max. 56 pts) #9.(max. 20 pts) #10.(max. 24 pts) Total(100 pts) Properties and Formulas for Laplace Transform 00 Let F(s) : 1:{ m} = [Doe—stf(t)dt and 0(3) : £{g(t)} = / WWW, In the follow
0 0 ing a and b are arbitrary constants and n is some positive integer. £{af+bg} = aF(s)+bG’(s) (1)
MN I 8F(8)wf(0) (2)
£{f”} : 82F(8)8f(0)“f'(0) (3) z: W} : sws) — Wm — s”'“2f’(0) —   e rimUm) (4)
Harm} = Raw) (5)
£{f(ta)u(tﬂa)} = 6‘”F(8) (6)
£{tf(t)} = F'(S) (7) £{f*9(t)} = F(S)G(8) (8)
mm} = 1 (9)
£{t"} = 3:11 (10) at _ 1
we } e H (n)
n! £{tneﬂi} = (s—a)n+1 £{cosbt} = sgibg (13)
£{Sinbt} = Sgibg (14)
at 1 _ 3—11 C(e CObbt} —— (s_a)2+b2 (15)
at; _ b £{6 bll’lbt} —— m VERSION 01 1. Multiple choice question. Enter your answer in the computer scantron.
Find the inverse Laplace of the following function: s — 7 __ 3"? __ _ S '— :f"
32 + 2.3 + 5 " p54— } 31%. ' ( 9H)"; 5'
(a) 6—” cos 2t + 2e_t sin 2:
, a
(b) e—r’cos 2t — 46"" sin 2t :: “git—g}: :1 a— 11L Q
(C) 284 cos 2t — 6"“6 sin 2t )‘f' {5+0r 2), {stiﬂiz/L
(d) 26“ cos 213+ 6" sin 2t L #3" (e) 4e“ cos 2t — e”t sin 2t — ~ )
e Cvmf~ 6% 9494*
2. Multiple choice question. Enter your answer in the computer scantron.
Find the inverse Laplace of the following function: 3+1
8(32 +1)
E23??? 5+) 2 i AM
(C) 1+ cost — sint SigEr} ) s ) 5+}: H9311)+(3§1“95 (e) 2 e cost+23int 3. Multiple choice question. Enter your answer in the computer scantron.
Find the solution of the following differential equation: y”(t) + 4:9’05) + 4W) = W), where g(t) is some given function. t (10t+4)e“2‘ +/0 9(tT)T€_2Td” +4[§\/_ + if: (1015 +4)th +/tg(t —T)T627 d7 2
0 (S'fLLQ—Lgyj)
43+ (8 1L G9 (a) W) (b) W) (C) W) t
(10t + age2t +/ 9(t _ 7W _ T)eg(t—T) Ch
0
t _.
‘— FE (18t + 4:)8—2t +/ g(t — T)Te”2" d7 0 (d) W) t
(1815 + 4)62‘ Jr] g(t — T)T€2T d7
0 (e) W) 4. Multiple choice question. Enter your answer in the computer scantron. Solve the following integraldifferential equation: 1 9505)   2 f0 (tw€)%(§)da:t, 99(0)=1 (a 497%? )
(b) cost ‘ Xx): + ((c) (310: 2t — sm)2t 3.143
d) E et — e"t
~ M 6/9
._ I 71. ——.‘n_____
i a a ) egg) @}@L
L g? g :3 3: 3W+ 6&3 .1 51— 1 ~Lr 32+ (ML (9—2)?”
33 _ Tgix SL' : 9 ID 6@
c ASJL_: j‘ y ﬂﬁ.¥%% ~»
48 H0 6 +&aﬁ 5. Multiple choice question. Enter your answer in the computer scantron. Solve the following differential equation: Wt) + 3y’(t) + iii/(t) = W), 31(0) 2 MW) = 0 where g(t) is the function which is equal to one for 1 S t S 2 and zero otherwise.
( 1 (a) (6” “1) H %6_2(t_1)) u(t — 1) — (Pill—2) — Ee"2”2)) u(t  2)
(b) (e—t —~ e_2t) u(t — 1) — (e’t — e'”) u(t — 2) (c) (8—0—1) w e_2(t_1))u(t— 1) — (is—(“2) — e'w’zl) u(t H 2) (d) ( w e—E + gig—2‘) u(t — 1) — e et+ $840116“ 2) * h l c—g .13
>19— S‘l$~+I)/J‘+2) (9 r 8 ) falg c ago/cw “ ‘i‘V‘ §7+ E12; /: ,4(m>(m)+ 5 (gym) + C (9H) S 6. Multiple choice question. Enter your answer in the computer scantron. Find the Laplace transform of the following function f: 7. Multiple choice question. Enter your answer in the computer scantron. Find the Laplace transform of the following function f: ﬂt) (sum of an inﬁnite sequence of delta functions location at 0, 1, 2, 3, 4, . . . and so forth)
1
a
( ) 1+ 6“ 79°”?— 6 we We mg we» ~~ ’ 1 8. Multiple choice question. Enter your answer in the computer scantron.
Apply the following ﬁrst order backward Euler scheme: yn+l : 971 + f(tn+1»yn+l)(tn+1 _ in) Consider uniform time stepping size: in“ —tn 2 h. Let yo = (9(0). The explicit formula ytfﬁwt to solve: for yg is: 2~3M
ﬂ+hF
1+h+m
b
( j (1 + h)?
__ 2
(C) 1 h +317,
1+h2
1+ 2h 2h? —2h3
(1 + h)2 (M (M 9. Find the Laplace Transforms of the following function: (a) 1526‘ sin 3t. (Hint: make use of properties of Laplace Wensform! There are many ways you can do this problem.) i
(b) 9(t) = / ﬁg) dﬁ. (Express the Laplace Transform of g in terms of that of f.)
0 (Hint: try differentiate!) r it 3 “Zn—Wm— 31'1/3‘9‘ é" a“ “PE—4 ‘gxwll/W) (310 + C7 10. Given the following system of differential equation: :i: = :c+ya:(:c2+y2)
:9 = —m+yy($2+’92) Consider the polar coordinate (736) representation of (3:,y), i.e. T2 = 3:2 + y2 and
2: tan’1 d,
(a) Find —r. Express your answer in terms of r and 6 only. dt d
(b) Find at}. Express your answer in terms of r and 6 only. (0) Plot in the some graph the trajectories of the solution of the system when (i)
(56(0),!!(0ll = (05,0); (ii) ($(0l,y(0l) = (1,0); (iii) (MUM/(0)) = (2,0): 1 ' CHAIN RULE, CHAIN RULE, CHAIN RULE...) (Hint: i tan—1Q) d6 :1+€2’ 11 ...
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This note was uploaded on 04/09/2012 for the course MA 303 taught by Professor Staff during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 Staff
 Differential Equations, Equations

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