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Unformatted text preview: Math 250 NAME:
Fall 2007
Exam 2 ID No: SECTION: This exam contains 10 questions on 10 pages (including this title page).
This exam is worth a total of 100 points. The exam is broken into two parts.
There are six multiple choice questions, each worth 5 points, and 4 partial
credit problems. To receive full credit for a partial credit problem all work
must be shown. When in doubt, ﬁll in the details. No notes, books or calculators may be used during the exam. Please, Box Your Final Answer (when possible). Multiple Choice Section 1. (5 points) Suppose the Wronskian of two functions f and g is W(f,g) = sin(a:). Which of the follwing statements is FALSE?
(a) f and g are linearly independent on any open interval.
(b) f and g are linearly independent on the interval (—g, g). i/(c) f and 9 can be solutions to a second order linear homogeneous dif
ferential equation on the interval (—g. g). (d) f and 9 can be solutions to a second order linear homogeneous dif
ferential equation on the interval (0, 7r). 2. (5 points) Which of the following is a suitable form for a particular
solution y(t) to the differential equation y” + 6y' + 93; = 21342"3t + 48"“ cos(t) + 56'3‘ sin(t) ? A,B, C, D below are constants. r1+ 6r 4. 7 :0
(a) Ate's‘ + Bra3‘ cos(t) + Ce‘3‘sin(t) (“4' 5 $.30
0 —3t —3t 3t,  r‘ " r1 ' ’ 3
(b) Ate + Be cos(t) + Ce sm(t)
3 1 624:
(c) (At2 + Bt)e‘3‘ + Cite3‘ cos(t) +Dte‘3‘ sin(t) ‘ , 31'” / (d) (A123 + Bt2)e‘3‘ + 06"“ cos(t) + De‘3‘ sin(t) l0 3. ( 5 points ) A spring is stretched L meters by mass of 4 kilograms. The
system is set in motion at time t = (l by an external force F(t) = sin(wt)
Newtons. Assume no damping and take 9 = 10m/secz. Then resonance will
occur when 4 0 4—10 :‘lzL. ”5—
\/ (a) w2L=10. Co 2 )K <4
(b)w2L=‘20. 0 7: —° ‘xF—E
(c) w2L=30. “5 3‘30 ‘=‘~ .‘2 (d) sz = 40. 4. (5 points) If
Y1=t. Y2=t+2e‘, Y3=t+e‘+e‘t
are solutions to a nonhomogeneous differential equation 3/” + p(t)y’ + W)?! = 9(t), where W) 75 0, then the general solution to this differential equation is
(a) c1t+ 02(t + 26‘) + c3(t + e‘ + e"), where cl, 02, 63 are any constants.
(b) c1t+ Cge‘ + (236", where c1, c2, (:3 are any constants.
(c) alt + c2(t + 2e‘) + t + et + e“: where (:1, c; are any constants. /(d) cle‘ + age“ + t, where (:1, c2 are any constants. 5. {5 points) A spring—mass system has mass 4kg and spring constant
9m/secz. What is the critical value of the damping constant 7 (the value for
which the system goes from underdamped to overdamped state)? (a)'y=4kg/sec “43‘1“. k1?
(b)7=6kg/Sec 4r7'4 Yr *? =0 'l‘M Mf‘“+‘4/‘n£
(‘7 7—,
(c)'y=9kg/sec X. " (Pu“l
=> ..
/(d)7=12kg/sec < Y“ “‘§=I'L 6. (5 points) Which of the following CANNOT be the graph of a solution
to a differential equation of the form mzr"+'ya:’+kx=0, m,7,k>0? b
x (a) x ( )
t t
x x
t
(o) t
(d
)l/ H:r Partial Credit Section 7. (15 points) Given that y1 = m2 is a solution to
2:231” — 2331/ + 2y = 0, a: > 0, (a) Find the second solution yg linearly independent of y1. Explain why yl
and y2 are linearly independent.  grit—'L‘jliL :0 x70
um: aura, 37 xii , / 1.  P(X)alx .i
w(‘3"1‘3¥)¢w(93,v3’i)5v3 2 Ceg :C‘efxﬂb’C / _ if
”V“ — C8 hx=C><1. ly‘l' 91M 3" 0m; («kccwly “Ad{’1 we; w(3‘171.)*0 (b) Find the general solution to this differential equation. Gum/”(L \3:c.x1+ clx’ c,’ (I M
5 8.(20 points) Consider the nonhomogeneous differential equation
y” — 4y"+ 5y = tea + 3. (a) Find two linearly independent solutions of the corresponding homoge
neous differential equation. $7M Char 2 ' rl'ciNs—ﬁo, r r = L Mo‘q—iz‘
V” \, 1, .. L ..
7. L =LiL
x=1, ﬁ‘l
2t
3,:eltm’c. ‘31=€,A\t. ‘3‘, y 1.x... AxelLP mm, M WI— “49C propmh‘cmdl arm/(5., 5940 (b) Find a particular solution of the nonhomogeneous equation. 0‘: T‘g Y(+)= (At+B)e Ni.
%V 1 x
CD Y’: Ae1t+ll4tH3let =(1At+ mime“? 0 Y”: 2A€Lt+1(lAt+A+IB)eZt_e(4At+¢A+¢(3) .. ‘t ‘
6) ‘31 (Mt WNW) ‘ (+61% 2M + AHBB + y (Ae+s)e‘t+i~c : telt+3
he, I“M449 +SC= teltlrg‘ 3
@ YR): te Jr}. 6
if AdlWP Mlonzo 6, 4m 5/172! (c) Find the general solution of the nonhomogeneous equation.  1t 7x ‘ 2e
Q’C.£ mt+ c1e wt + ta + 3
E) CUCIMI (d) Give the form of a particular solution to the equation
4. y” — 43/ + 5y = 3e2‘ cos(t). Do NOT solve for it. Yes)" At game + (Steumgt, mo wt. 9. (20 points) A mass of 2kg stretches a spring 100m. The mass is pulled
down 200m from the equilibrium position and then released with downward
initial veolicity 2m/sec. Ignore air resistance and take g = 10m/sec2.
g (a) Write down the diﬂerential equation governing the motion of the mass.
714:1, L:o\(. 2(0‘—4L~0.i .L‘cZDO. LUk): F°s€l17 0“: rte W Rm QWCi‘Lviuva. after txmo 7’ 1. Differential Equation: 7. u." +2 mm = O 2. Initial Conditions: “(0) = o. 2 , u’to) = 7. (b) Determine the position of the mass at any time t.
0‘2E : iv — 7—(0
i. ha 1 1  I‘m: .
MngAAiOt' W1=M¢0t
H: C\(L.A(o~{: + C14A1~l°t u'= ~IO€,A/Cmiot + Ioclcwtot. \l 2 C 0 7.
u(o),C 0}, (at/(MTlOCL ' . 
one): 0‘; le0t 4 o‘LAQIOt 7%? b (c) Find the amplitude, frequency, period, and the phase of the motion.
(Macon, Cmtot + 0,7. xvcwlot = R M (”f ‘ 3‘) , “Li—Lu. m R '5 k0\z>l4_(0\1)‘ ‘2 (Klfl = 'amPitl'kde. it
_ 1‘
47¢». Zl X’QV‘C‘ijl‘z: szw
W
i 1. Amplitude: 0. Z J:
{ 2. Frequency: [ o
I 3. Period: %
.77 4. Phase: 1
q.
((1) Find the ﬁrst time the mass crosses the equilibrium position.
gut“; aim=0. rot—5:13 tot=§+% =23 MS fi‘gzrg/Lec, ‘10. (15 points) Find the Laplace transform of the function . ' 5 if05t<6,
f(t)—{t—1ift26
from deﬁnition. 5
0° e—Atdt 2p
_ : 4:
(:06) iéiw} L “C”
A ﬂ‘t‘
 6 ~a4t 1‘»... 1 e pit gr];
“SOS—e. 0H? +A’I“ $6 U; > f
“—454 ON: e At
~ “At 6 dc“ alt we
: A. Q 47/ ’ ’U’ = Q
”A 0 €223, ‘A
J‘t’ A A _ ’t
’F LE“ 0:") .6 4" g e ’6 At
A—vso ’ﬂ 6 +,4
 , éA — ~6 _
se i+l9t(/*">QM_§.€24+?”k
2»? V7: 3: A“ ‘A “A 241 e
‘ J,
: \_. g 65, g 0 A~A>O
T “L: 4. yew \ er” 64
R + Lb» R e
,4 A—w ‘Aa. — V
J.
0 FL
'3 : + Q‘éﬂ A>°
' Pi A :— ' ”7"“
Hrs)? 8 )7M; ...
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 '07
 GYRYA,VITALIY
 Equations

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