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Unformatted text preview: i" x“? if 2 5’ ‘5’; ? = my wa 2 G§§z§ff§§$ Q??? gﬁw'ﬁkaﬁ ﬁf’é‘é/ W *5/55; Va ﬁwﬁ/ﬁ w the general solutions of the following differential equations: a) 5” i ) ﬁgmyﬂxl) da: 3(4 ~— 3;) Answer: QA//w xixv/ZthY/fc 2 3 2
b) _ (sec2m+3:52tanyw;y3—) dx+ (x3secgy+4y3+%) dymo
U \A’ W Answer: WWW; ' O m m.»  A bail} attached ’50 a spring is suspended in a large bucket of water. When the bail is
displaced from its equilibrium position, the equation of motion is given by mu”+4u’+7um G,
where u z u(t) is the displacement from the equilibrium position, and m is the mass of the ball. a) What is the largest mass m for which the hell does not osciilate (bounce back and forth inﬁniteiy often)? m 7””; ifff71'0 f“:  ‘f'fV/ngé’m ﬂ 2/11 ~#7 [#7714 330 (=3? 77? 5% Answer: )2 <>
Qjﬁ (continued) 13) Suppose that m is large enough so that the bail osciilates. Find a. formula for the time T Between successivelocal maxima of the displacement Mt) (1.9., if u(t) attains a local maximum at timet m 81
and the next time it attains a local maximum is at time t = #2, ﬁnd T 2 t2 «— t1.) x, yea a £9 {ﬁg/I 15+ Ur“ :_ A r 7/, _
.géz‘é’zi 4;», 3%me 2‘: éz’kwczwm,
‘ .a Z , ML :52... "1. ‘ ﬁt
‘ ‘ 2 27'»: , ‘
Answer: T = 31$ a”; I differential equation y”—i~:ey’+29= 0
can be solved by power series (about the point x0 m 0). *6 . #v’
a) Find the recurrence relation: y: a", 2’ w y /~ grid?“ I
5i“) q I
y//: no T’ﬂ71tz//ﬂ‘é//41{fp I {w ﬂﬂ/KM/ﬂm */ W 1/ 3"": &
Mew mi: 0 We) an Answer Qﬁh x 72:7 (77 2 b) Find the ﬁrs’c 6 terms of the series solution with initial conditions y(0) = 3, y’(0) m ml. 5?
QL3’7€'#?
a 1..
Q’:‘J51p
Z
“ﬁn—g]
4 ~ “3—7;”?
5" 94 d” lﬁlnswer:y=i 3 x2+$3+$4+15+~ (Numerical values for the coefﬁcients should be placed in the boxes.) '  ' The diﬁez‘entiai equation 2:52y"+$y’—(x+1)y20 I has a regular singular point at a: m 8, and the roots of its indicial equation are‘ and ~1/2. Fill in
the boxes (with real numbers) in the foilowing series solution corresponding to the root “1 / 2: T 1
W —1/2 4“ “ad 2 a” 3 H.
4 _ H a"? w 1
Y" :aﬂx 1 610' // (a~:*4_z‘“~:“OJ yr: [@J/aLIW‘
M: ng~%}(/I'§']4¢ 1’” Va . ...,l
2 17%.: 2 @aé/zzz—KM X” ﬂ g%
33/ :2: Eétviz/Qx Z 2 Mi;
=5 ~‘" 5?“ 1’
wry « E—a,bft’”: "‘ ’
w/ : "Sadfﬂvt
o F. ,, q}, £[ﬂ%}(Zﬂ*77)”7]“ airr “a
W
ﬂmﬂﬂp’)”,
zmw5ﬁ"/"/ 72627732
Qh‘i" 27’"
IzZZZIij ’7 7
I _. Ml“;
I?“ 0’1 9'"
1 «.11. .5 amide; the “sawtooth” function 1 I
m ' ._...< __
f(:z:) :cfor 2mx<2, Fill in the boxes (with real numbers) in the following Fourier series expansion of f : ‘ :c a + cos 2m: + /’ sin 2m: + cos 47m
. I I ' + sin47rcc + [E] cosﬁvrx + smﬁwm + It f(:1:+ 1) for all 2:. W .. r. Consider an elastic string of length 5 whose ends are held ﬁxed. Suppose that a a: 2 so
that the displacement Mm, t) satisﬁes the wave equation 41:,“ 2 at; , u(0,t) 2 145,12) 2 O .
At time t :2 0, the'string is set in motion from its equilibrium position with an initial velocity
mm, 0) 2 sin 31m . Find the displacement Mat, 25). co " . h m .._./
1’, “I! ——d_.~"" \f W,
. I1 I
'7 77’ _ nlﬂm ___,,
“ﬁzz/(h 3;” may»? 5..
3maa (4Q); E 12;}; h M”;
()5)
\ “fr’
"a ,
13» fiefd IJ 33¢ aye fey/{1 Mil/4
I 3/77
~ a,“ .__/_
Z l“ éﬂ’k“.
\ .1,
[(13,677
N (7950 Agni/5’) — .1... m" ,Eﬁ’fmyﬂ“?
Answer2 Mat) 2 l 6??" £01 0775,05 7‘ ( ."':&:'w».r,==h;...; :1) Find the solution of the linear system: dx
{3%} : 7$1+6$2 d3: _
2&3 :3» 2$1+6$2 Hint: The roots of the characteristic equation are r; = 3, r2 m 10‘ ' (f) __ 4: 213 5 'ﬂ/ S
g y. ' a 7 V " 3/] a {—3][5 0
4% 2‘63: = 0 m3 *5??? xﬂ): EQf'Zé:
V“): 4L ?—/D 6 ’ﬂz] : ~31! +531: 0 ﬂzﬂng/ $37 ’1' {4/5 6 227’! . ‘
{rm/2‘; 91/”
_ {714 ' [05 73‘ (95‘
Answer *Lryfé “’1' 3, $2.: “QC/e" {mite
. 13) Find $205) if the system has initial conditions 331(0) 2 7, 332(0) 2 0. ycﬂzcz: ,2
“‘24; 7:61.. J. a fan?
Answer: zit} : “' 2 6331‘ 7“ (Note: 222(6) is a function, not a vector.) Part II. Multiple Choice (30 points) Circle the letter corresponding to the best answer to each of the following problems. You need not show
your work on the multiple choice problems. No partial credit will be given. For each of the following boundary value problems, What is the minimum value of A that gives 3.
solution (other than the trivial solution 3,: 2 G)? ABCDEFGHIJKLMN®Q i. y”+)\y=0, y(0)m0, y(l)=
ABCDEFGHIJKLMNPQ :2. enemies, y’(0)=0, y(7?)=0
A. —~2 B. —1 o. —~1/2 o. e E. 1/4 F. 1/2 G. 1 Hi 2 I. 3 .l. 4 K. 7r/2 L. 7? M. 271’ 712/4 P. 7.”? Q. 47:2 B C D E F 3. The differential equation Et— 2 m5(1—y)(l~§)y has 2 asymptotically stable equilibrium solutions and i unstable equilibrium solution.
B. has 1 asymptotically stable equilibrium solution and 2 unstable equilibrium solutions.
0. has 3 asymptotically stable equilibrium solutions. D. has 3 unstable equilibrium solutions. E. has no equilibrium solutions. F. none of the above. Which of the following statements best describes each of the following differential equations? A B C D 4. x2y”+27y’+(x29)y:0 B C D E F 5. (<:os:3}y”+xy’+2ym0 6. 2531;” + (551:4 e ﬂy’ + (2x w 3)y m {3 A The point m 0 is an ordinary point. B. It has a solution of the form sf} + a.) + (L223 + {132:2 + . . ..
C. It has a. solution of the form 334/2 + ago”? + (223.7ng + 0.35532 + . . ..
D it has a solution of the form 331/2 we (21:53” + a2$5/2 e owl/2 + . . ..
E It has a solution of the form :52 me (113:3 + (1,2324 + (133235 e . . .. F None of the above. ‘Whieh picture bee: describes the type of each of the folio“?ng critical points? dm/dt : y w :53
A B C D E F G H} I J T. the critical point (1, 1) of
‘ dy/dt m 21 —~ div/alt = y A B C D 23 5F G H I J 8. the critical point (0,0) of
dy/dt = —25'1n$m 4y
aim/(it m “33: + y A B D E F G H I J 9. the criticai point (0,0) of
" " dy/dt == W373: + 33;; (tr/(125:: 15(20 — :1: w y) A B C D G H I J 10. the criticaipoiné (1515) of
dy/dt : y(30 — m w 3y) sink source  sink source 
A B C D %
sink source sink source ...
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This note was uploaded on 11/11/2009 for the course MATH 110.302 taught by Professor Brown during the Fall '08 term at Johns Hopkins.
 Fall '08
 BROWN
 Equations

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