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Unformatted text preview: Alex Boismrt Math 3313., Winter ’03 February 29, 2008, 8:00 AM Midterm 2
Name: __ _
Student ID: \/ I?
Signature: __
Check your section: _ 1a (Tu) _ 1b (Th) TA: Neel Timviluarnala
_ 1c _ 1d (Th) TA: Eric Radke This is a closedbook exam. Do not use notes, books, papers, or electronic devices of
any kind. Do all work on the sheets provided. Do not use your own paper or blue books. If you need more space for your solution, use the back of each page; you may request
extra paper. Be sure to state clearly if you are continuing on a different page and label
the problems well. Do all 5 problems. For full credit, you must Show all your work. Do not. worry about
oversimpljfying your answers. Please clearly indicate your ﬁnal answer, for example by
putting a box around it. Problem Out of Points
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2 9
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5 12 (1) (9 points) For each of parts (a)(e), determine whether or not the statement is
true or false. If true, Show why. If false, explain why not. (a) (3 points) There exists a differential equation of the form it)" + MW + G‘th = 0 such that y1(t) = «3t and yﬂt) 2 t2 are two solutions
on the interval (—10, 10). (kids/0w OM; (M exei; Mr LOFDQLZ‘QA WEIde
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boil M We: (b) (3 points) If A and v are deﬁned as we we) then v is an eigenvector of A with associated eigenvalue 2. “Ma—To awarigmtiomwn
(Av: LE)(?>:(J:3:&(;):;M (c) (3 points) The differential equation giving the following direction ﬁeld is autonomous. .* § Myrrh . Ehxsssaviiniflanﬂﬁﬁi .
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"Jx\»92 “HA; QVdeﬂngo/x W (SLAVE 1\/\ VQ. NW (2) (9 points) In this problem we consider the differential equation :12’ = x3 — 43:. (a) (6 points) Draw a phase line for this autonomous differential equation.
Classify each equilibrium point as either unstable or asymptotically stable. £00: xii—letﬂ'ﬁﬂ: XCX~2Y><+2> (b) (3 points) Sketch some solution curves in the tm—plane, indicating the
equiiibrium solutions. ‘// ///'
\\\ @\ a ,j (3) (12 points) A 2 kg mass, when attached to a certain spring, stretches the spring
20 cm. Assume this system is unforced. (a) (3 points) Calculate the spring. constant. for this Spring. F =mgzl<x so seem02 :ekzss g (b) (3 points) Find the damping constant ,u for which there is critical damping.
cggnalj/{lalf4 n r” ,ﬁ r x
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s”? 90% @1390 C: ﬂéiﬁz¥rCﬂo E. (c) (6 poirits) Find the solution to this system given the initial conditions = 1 In? y’(U) = 2 m/s. mm mm :0
g; a :1 l
ymwcrno ==~>Q+¥330 =7 A“? (4) [8 points) Find a particular solution to the differential equation
1;” + 93; = 'sec(3t)
(Hint: Recall that ftan(t)dt =1nsec(t) +0) LL12. we umbu'hm“ {39‘ PWW
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W 6% 3% £05( 39 *gvzf 9M 3%} 2 ged 31:3 \JJ EBA/«(356$ + \lzf CDSC3E\ DO (5] (12 points) (a) (4 points) Prove that the imaginary part of the solution of 2” +2" + z = to“
is a solution of y” + y' + y = tsinbt). (Hint: let z(t) = $(t) + iy(t)) (b) (8 points) Use this idea to ﬁnd a particular solution of y” + y' + y = tsin‘(t).
(Hint: solve z” + z' + z : is“ using the method of undetermined coefﬁcients.) let9405+ =3 9W9 : {statute ere: (tHUeH'
g0 w \mr «Ca a 90mm ya omega/x (we) a“:
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This note was uploaded on 04/02/2008 for the course MATH 33B taught by Professor Staff during the Winter '07 term at UCLA.
 Winter '07
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