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Unformatted text preview: M427K Exam 1 , February 21, 2006 Question 1 [10 Points] A 1000 L tank initially holds 200 L of solution of the extremely toxic Uranium Nitrate
salt U 02(N O3)2 with an initial concentration of 2 ,ug/ L. Solution containing 8 pg/ L of U 02(N 03)2 salt enters the tank at the rate of 4L/min. Well mixed solution is
allowed to ﬂowout of the tank at the rate of 2 L/min. 1. [3 Points] Write an initial value problem describing the amount (in ug) of
U 02(N 03)2 salt in solution after t minutes. ‘ , C\ d? _ A GLO
gig: Rmxcm‘ Rom“ ea 3%C5X4 1 1006“: (Mel =4OO 2. [5 Points] Solve the initial value problem to ﬁnd the amount of U 02(N O3)2
salt in solution after t minutes . Sill. gilﬁl as 151, [00% i at ‘ l
d_Q EL: * 'DZOOit Gd =ln too+t
O‘NW 31 PL 1 [y SO REC): IOO’rt ji Qoma Q] = mow)
[Omag = $100+, + [6&2 +C
ra= W 9(0):; =3 C=4oooo
C9 lOCH: \OO _ ltt1+3zoot #0000
QC» Moo 3. [2 Points] Find the moment that the tank is full. Find the theoretical limit—
ing concentration that would result if the tank had inﬁnite volume. Will the
concentration at the moment that the tank is full be lower or higher than the
theoretical limiting concentration ? You do not need to compute the concentration at the moment the tank is full. The tank I‘S [Ni oi t=400 WW8. l? H412 [QM hod m ﬁnite Volume
and w [d at time 90 lo +oo Na would have 0 lwmimg 
Conmnir‘olion oi Climilmg = Cm 1 8 93/1. ‘ HM ; "M concaniroxlion 1‘) ~[own W4 #14 limihn
Stimulmiglnu Ne NIL hot/2 [M conwnirqiion (ﬁrth?
WOW/Li [We WW [S [W bang William the Womb ch[ 2 [MW ContamimilOl/L M427K Exam 1 . ' February 21, 2006 Question 2 [10 Points] 1. [3 Points] Apply the Existence and Uniqueness theorem for ﬁrst order linear
ordinary differential equations to ty’ + y = tsin(t), WT) = 310 giving the largest interval for which solutiOns are guaranteed to exist and be
unique. ’ i. Pitt (19‘) (1r . , h '
in standard Perm 8 + z 5 #smt. yti) and q ) are (on mom on Hue whole. real line excluding JC:O ler‘e pill hob 0
Wm mgmpluh. in large} MiHVCti animal/2:43 it loud QXCibLOimg
V‘Q is COME) i Q)? giomimd ﬁrm i 2. [4 Points] Find the general solution of the ordinary differential equation
. ty'+y=tsin(t) , . Art) = e‘"’“=L
0% [t8] =t5mi '
£3 =ft3midi
; «Most + it“ OR
., ~tcori + sm‘r +0
._ i+ m + 0'
@ W ‘ e t 3. [3 Points] Find the value of yo such that the solutiOn of the initial value problem
ty’ + y = 15811105), . y(7r) = yo exists for all times t. Explain What happens for other values of yo. 3: —C0_Si;+ int {9 OLQRYIQOi For 0“ t This gives BCWF—‘cosurh 3%“ :l :3 90:},
\9 96M ihcn (>6 and We solqhon 8012) )0 +00 068w
(upfrooch ’90. 8mm the right > g '
l? akal "Mm 'CLO and HQ >olul10n3t>m is ~00 q) 301:? avr'wd‘ W “0"“ M “3” M427K . Exam 1 _ February 21, 2006 Question 3 [10 points] 1. [8 Points] Find an explicit solution to the ordinary differential equation @_4t3+1 dt 231—2 Sayorqble 533014 = {ffH, +C
Q54? : t4+tiC.‘ ~l ziw ‘j —  iJJL4+JL+C 2. [2 Points] Find an explicit solution to the initial value problem dy_4t3+1
dt _ 2y—2’ if 1— th+t+4 21(0) = —1 M427K Exam 1 February 21, 2006
Question 4 [10 points] 1. [5 Points] Show that the ﬁrst order ordinary differential equation
(4:103 + 33/) doc + (4y3 + 3:13) dy = O is exact and hence ﬁnd the general solution] You are nOt required to ﬁnd the solution explicitly. mil‘f‘th SOexocl. ,
Cling? [email protected]+?>3) dx = X4 +318 + (13)
Cbbﬁij) 3 %8+3X)d3 —: a!“ lgxﬂ + DCX) erﬁmaweoﬁwr\'¢®8>=X““asrg* Or)
and in general solubon u _ xq+3x3 +39 = C. 2. [5 Points] Find an integrating factor that makes the ﬁrst order ordinary differ— ential equation
(a: — yz)‘dm + 2mg dy = 0 into an exact equation. You are not required to ﬁnd the solution. MSNL=4g M: _’_l_ » defend) onlj on x180. can ﬁnd 1
N x depending 0an on x C.“ M427K Exam 1 ' . February 21, 2006 Question 5 [10 points] y Solve the homogeneous ﬁrst order ordinary differential equation dy _ x3+y3
E 3my2 '
Give the solution explicitly if you can.
3
ﬂ : High)
0W 305/1?
Lei V= 3/x 36
HP
X3311 +V = 151%
ﬂ_ W” _
7C oix _ ”5‘41 V
= W” 4%? .
3V1 3v7’
= 1““ V?’
V2
3‘3"" = d><
Hv?’ X
Z “214$
.3V 3 dv do _. vex/10W
i'lV “lidu .—. 5V1dV
g "id” : ~‘ In lot} = “dilly“ i'lvgi
M
fl ax: lnlxi+C
‘— .n\l~lv°’i= M M +0
7. . '
mung] = \n Ail
)“1V8 2 1' ADC—1
1 I ‘1
V3 = I 1 A1
3 z ‘92:)CgiA/x M427K Exam 1 February 21, 2006 Question 6 [10 points] Consider the autonomous ordinary differential equation E = NJ) where f(y) = y4(y2 f 4)(y2 + 1)(y — 1)3 1. [4 Points] Sketch f (y) vs y.
Roois al 5=O 3:1?» Onol 3" W 2. [4 Points] Find the equilibrium solutions and classify each according to its
stability. .
Equilibrium solubom ore _ 3: 5L unsiablt
‘5 3 O semisiable
' ' 5 = l osgmpluhcollﬂ Slﬂbl€_
3 a l uHsloble 3. [2 Points] Sketch y(t) vs t. ...
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 Spring '07
 Windsor
 Calculus

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