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Unformatted text preview: 10’pts ' (1) Find all solutions to the following initial value problem. y should be mpressed as an explicit function oft.
ty’ + 22527,: — 47546":2 = 0, 11(0) 2 4 @011: y’ +21}; : 4439'“?
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9% ”044w W :4 ' Q04 : OH, {OO 3 (2) Find all solutions to the following initial value problems. 3/
should be expressed as an explicit function oft. , 3t2+1_ __
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E 10 pts (3) Find all solutions to the following initial value problem. y may
be expressed as an implicit function of 9:. 2x + ycos(a:y)
I —~————— = 0 1 = 1.
+ m.cos(a:y) + 33/2 ’ y( ) I : _ 2X+Y694pr
\I XLoley)+%\r .lxwﬁesmy) + [WEXYWWJ y’ .10 Lp ; x‘ + QTMXy) +£er ow ‘ , 
W " XMW) l “05 “W ‘38; Um =3} if» : x‘a— Qa‘anys + %3
are/neml Lolﬂl‘é‘mi  xlE anxy>+y3;{/
n'yu/‘el l+ EPA/EM :l‘, L: NEE/w v u <\\\\\\1«W&4¥wmw\mmwzmww«rmwwﬁw\ mm» a“ N swwxm mwm “ 5 (4) Find the general solution to the following differential equation.
3,] may be expressed as an implicit function of w. 2
xyy’—a:2 ~y2e (x) =0. XH= W1 95%;;
y) : Xi+ +!i€vl)
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+5159 —L) y: She 12 pts 2 \DO (5) Multiplication of both sides of the following differential equa—
tion by y“ for an appropriate value of a converts it to a type
differential equation that we have studied: (3:264 +5y2’)y +4x3y=0==0. What is the appropriate value of a and what is the type of
differential equation we have in mind? Prove your answer.
4 pts Do not solve the resulting differential equation. 4X$y yd + xgx4ya+xyl+d> \1/ :19 ‘ AW WW hm side/s (>5; 31 we Ga (6) The substitution 1) = y3 converts the differential equation
33:2y’ + my = 8””
into an equation equivalent with 7/ +p(g:)v = Q(x)v". ‘  Find 10(90), q(m), and n. , , 6 pts
y’ ; é” V~§ v’ xv’+ xv 23H?
, i. ‘9} .3
v + xV ~ ‘V
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D. memmwmmwmmmn“ warmmxsmmwmwmymmwuwmmwxﬁwsmmxavsmmwmcr:wawmmwwmmwmmwmxmmxﬁwwﬁmﬁmﬂ 2 pts ‘4 pts (7) Consider the initial value problem (em '1)2y’ey3 = o, y<b> « 1. (a) Find a value I) such that the either the existence or the
uniqueness (or both) of the solution is not guaranteed
by the fundamental existence and uniqueness theorems.
Justify yourX answer using these theorems funny 1 W 15w. {toque uolm @oM'M'dwlso ‘5 X «“1032 go new l? 0,11%le Nib/merm
M%11W5 (trterm): CA wet QWQM ‘ (b) Attempt to solve the given initial value value problem for
the value of b you found in part (a). Is there really a
problem with either existence or uniqueness? Explain. Hint: f(e”c ~ l)”2e“c dx = ——(e”‘ —1)“1 + C , 21. 22
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“a ‘ (299—! M3 “ ‘ (8) Consider the initial value problem x+1’y(0):b' (a) Find, a value b such‘that the either the existence or the
uniqueness (or both) of the solution is not guaranteed
by the fundamental existence and uniqueness theorems. Justifyg our answer using these theorems. 2 pts 4 “E \ .
ﬁlm a \J’ AALL tk 4 .i ow“:
y> X“ 3%; imxli > r T was? 3» =4, QM 95% Cl AXAWWM ad" v.14) go
bit WWW bl‘ W myeaea; UM 52 0V (b) Attempt to solve the given initial value value problem for
the value of b you found in part (a). Is there really a problem with either existence or uniqueness? Explain: 4 pts
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with WW,
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y l0) .: 4, C10 {VAC} bud‘%§/L?/%Q 44$. \ZP’Dllé, . (9) Consider the following initial value problem. Without solv—
ing it answer the following questions: 502—1
’~ 21(1):? _ 1+$2y2’ (a) Is there just one function y(a: )satisfying this initial value
problem? Explain. _._y_...__, ”9% Sim9 399V “ll—7W) “Ml h§$w WEQ 40W (in, 34 My iv the aiﬁemw WWWMQ
amen l0 ka Grivewe gelmmwww (b) On the basis of the fundamental existence and uniqueness
theorem what can you say about the set of values 33 for
which y)(x is deﬁned? 4.. 5 m 9/ 91M, Wexfw sow) alwmelwmy .11 into/WE» l Wish/(fix
WEE M9 max ATWl/d‘rjv \/L)<>\J mi WNW. (c) Assume that the solution exists for all :1}. For which values
of m is y(m ) an increasing function? y HXR/ >0 is”. 0%) 3 PQM 5% WWW « 11
(10) A pond initially contains 106 gal of e h te bWater con—
voy—0‘0! SM It 1% taining .03 + .01 sint lb/gal of pollut nts ows into the pond ‘ at 13,000 gal/day and ﬂows out at the same rate. In addition, ‘3 9'5 9 a“! Mha‘ ‘  ere is a house next to the pond that dumps 2 lbs of pollutants . J p the pond per day. Find an initial value problem (differ 1‘” ”“6 ential equation and initial condition) that could be solved to ﬁnd the pounds of salt Q(t) in the pond at time t. You may
assume that the pollutants from the house are dumped at a constant rate and do not materially affect the volume of the
pond. : Mo + gotta? +1 V» OLOB 52%} E
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l 8 pts 12 T Penna “t (11) Assume that an object of mass m kgs is thrown straight up from ground level with an initial velocity of 4m/sec. As
sume that air resistance produces a force of magnitude FW ~ .0001——+—L— Where v— —. u(t ) is the velocity at time t and :1: — :ct( ) is the height above ground at time t. . (a) Write a second order differential equation. that could
be‘solved to ﬁnd a formula for x(t) during the period
the object is rising. Your diﬁerential equation should be
totally in terms of x; no 12’s allowed. Do not solve the
differential equation. 4" V3 \ 5
Mao; e l: 3W #0 1900‘ TH" \ V vs Pesgttve‘) (b) What are the appropriate initial conditions for the prob—
lem 111 part? We) = 4 Owe/t [/f
X 110 )Write a econd order differential equation that could
)be solved to ﬁnd a formula for x(t ) during the period the object is falling. Your dijj’erentz‘al equation should be totally in terms of 5c; no u ’3 allowed. Do not solve the
differential equation. W\.Q\l F ‘1"— "OQEQ in“;
if)
0“: “"3 ~ 0 00.91wa; ...... ...
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 Spring '08
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