Midterm-I-Solutions Fall 2010

# Midterm-I-Solutions Fall 2010 - Math 240 Midterm I October...

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Unformatted text preview: Math 240 Midterm I October 7, 2010 Professor Ziller Name: Signature: TA: Recitation Day and Time: You need to show all of your work. A correct answer with no work will get 0 points. If you see a shortcut, you need to explain it. Please circle the answer for each problem. Each problem is worth 10 points. It is more important to do the problems well that you know how to do, than it is to ﬁnish the whole exam. (Do not ﬁll these in; they are for grading purposes only.) Total Math 240 Midterm I Exam February 19, 2008 Name: 1. Find the solution of y” — 3y’ + 2y = 10 sin(a:) with y(0) = 1, y’(0) = 0. mm TL *33/121: (rvlﬂrﬂto to! I, 2x x VfQQ +CZ€ yp‘A/mﬂqv—Bwk \ Iv wa ~13WX p Y” “~A/m'k 4ng P Y” 5‘/‘+17: «W443 5/3: +028)+WX('A héLSrZA) :10/M2ux ’5ng :0 )023:39 g5 A, 315+fvﬁii) L}; : WX +3mx y: Clezx +Cz€k '+-/I/I/;"\X Q,wa W): cl“:Z +5” _: O \//{D):ZC( 'ch‘f/I M—“WT—"f 2x A I M X {—5 MOX \/= 12/ ’5 Q “7’ 4 \1 M Math 240 Midterm I Exam February 19, 2008 Name:— 3 2. Consider the linear system of equations Ax = b and let M be the row echelon form of the augmented matrix (A | b). In each of the cases below, (a) Bring the augmented matrix M into its reduced row echelon form. (b) Identify the set of solutions of Ax = b as the empty set, a point, a line, or a 3—space. 10132 12121 (I)(00121> (11)(00020) 00000 00112 1 L CD13 0! I 2 “M 0 {90016 X 57%“ was yﬂ: X:"S .E'tigz‘f " a ' , 794W /M’V474/ﬂ4 W W w’) A, we W - 22 1 ( <— J“ {9/ - I V7 ‘56) i l 1 O O U \[pO Oi O \0; o d I H2. 0 0 -—I w/ a} O 1 \00 a (D 0 )<:..,i 3‘2 0/20 YZSij:\1.-\ZS {1% ln/To Math 240 Midterm I Exam February 19, 2008 NaIne:____—_ 4 3. Find a second order linear homogeneous differential equation where y1 = 62\$ and y2 = \$62”5 are solutions. Math 240 Midterm I Exam February 19, 2008 Name:—— 5 4. Consider the following system of equations: \$+y+22=0 m+2y+52=0 x+y+(k+3)z=0 (a) For what values of k does there exist a non-zero solution? (b) Let k be the value determined in (a). On how many arbitrary parameters do the solutions depend? I (3‘ Hz? 0 “533gng (H4912 %? W 0 M Math 240 Midterm I Exam February 19, 2008 Name:_—___ 6 5. A spring satisﬁes the differential equation y” + 4y = 0 With initial conditions y(0) = x/2—, y’(0) = 2V?- (a) What is the amplitude and period of the motion? (a) At What time does it reach its equilibrium position for the ﬁrst time? (b) Draw the graph of the motion. 714,4:0 ,nr—ztlé V10): Ci’ﬁ/ V/’°‘):ZC’L-:Zﬁ W; Wﬁz Amwawl/Mmlxél {2 f; : ALrgb/wgtdé-I-J) Mir? / 7/ 5‘11 W52i—9 5_ E?) l V??? on - z 7? \3 W5: '7/AZHB7’ Z7T‘_/ a) WWW :1 W451 F ‘3: I“ siir ‘l§§.l* iii-w. Math 240 Midterm I Exam February 19, 2008 2 6. Find the solution of y” = 3% with y(1) = 1 and y’(1) = 2. “2 a X 7 IlyZO wa/ .2,- x/ixr my-” ,1 : r -r #2 : Lrﬁglﬂrﬂ) TC 41 7” :—~[ Name: Math 240 Midterm I Exam February 197 2008 Name:—___ 8 7. Let y(a:) be the solution of y” = :v + y that passes through the origin and has a horizontal tangent line there. What is the value of limp;Do \$612? V57 xx 4 a ‘ 2— ,0 Triﬂ 7/? ark (ML-y {/«v ~\ :: 7 /) >< —»< Z94: LIE, haze) /' 1.x y 3 \7 "I. :30 7° 4 VP 42%: >< A=-1 Bro yr, = xx »X \/ C {‘C’lez {A . I . \/(D) = 0 y /o):D We): cl +C2 2 o CX+C.L:D lcft (X: r ‘ '\ ‘ ,_ .: L y{:)>: C4~Cla( :L/ Cl—CZ_( Z L_ —K >/ :: W X ‘; f: _ r l _O / MA / - Math 240 Midterm I Exam February 19, 2008 Name: 8. Find the general solution of y” = 4y’ — 4y + 62\$. I‘I/ { ‘LK \/ ~‘r\/ +‘r‘/ :€ .2 1 1X C _ Z X l/A : Lie f ZXQ . x , yo "; AKL Q2 wma e ‘ 1? y]: ZAxeszp 42/45/36 I ' . ‘er 1 CK \/ 1’: Zﬂﬁw .(. éﬁy 62% +‘L%A”X~€/ +r (ll/4X ﬂ, '7 w< A V ~Lp/urén/ ; X26 V574 +42%) + Xef’xﬁf/J: ~S7/~U 1 42/45:“ EZAQK:~C A; 2K ZIK _L ‘2 Z./( Via-e +Cg<x +ZKC Math 240 Midterm I Exam. February 19, 2008 Name:______ 10 1 9. You are given that y = x“ e“: is a particular solution of the inhomogeneous diﬁerential equation 2 y”_2yl+y:_e\$ Find the solution with y(1) = e, z 1. 1/ _ (i I Y1 v ~27 +7” ﬁx 7/ YA / 1— , :; 1' W T ~ZV if! ' (r H Tl L x -: K- 42 x a A K K (ex st : L \ : C6 + 3Q K6 ’L b V / 4+4) 1 Z X : C17LCZ:0 Name: Math 240 Midterm I Exam February 19, 2008 11 10. This is the only problem where no work needs to be shown. Each answer is worth one point. Which one of the following statements are true or false: (a) The solution y(t) of a harmonic motion is periodic, True® i.e y(t + L) : 3105) for some L and all t. (b) The solution y(t) of a damped harmonic motion is periodic, TrueO i.e y(t + L) = y(t) for some L and all t. (c) If yl and y2 are solution of an inhomogeneous linear differential equation, then yl — 3/2 is a solution of the corresponding homogeneous differential equation. (d) If y(t) is the solution of a critically damped harmonic motion, then y = 0 has a unique solution. (6) For any m x n matrix A, the matrix AAT is symmetric (f) A homogeneous system Ax = 0 of 3 equations in 4 unknowns always has non—zero solutions. (9) For a homogeneous differential equation y” + cy' + kg = 0, with c > 0, y(t) = 00 is not possible. (h) If A is a 2 X 2 matrix with A2 = 0, then A = 0. TrueO If y(t) is the solution of a damped harmonic motion, Trueg then tlim y(t) = 0. (j) If y” + 9y 2 sin(at) is the differential equation of a forced TrueO spring, then the value of a under which one has resonance is 9. FalseO Falseﬂ FalseQ False® FalseO FalseO FalseO False® FalseO FaiseQ’ Math 240 Midterm I Exam 11. ml February 19, 2008 Name:—___..___ 12 Extra Credit Problem. You should not attempt this problem unless you were able to ﬁnish all other problems to your own satisfaction. The score on this problem will not effect the curve, but will be used as an extra consideration in assigning your grade at the end of the semester. There is no partial credit on this problem whatsoever! A spring satisﬁes the differential equation 4y” + ky’ + y = 0. If the spring is critically damped and a weight is attached to the spring is released 2 In above the equilibrium position, at a downward speed of A m/sec (assuming that A > 0). Determine all the possible values of A for which lim,g_,0° y(t) = 0, but the weight never reaches the equilibrium position. Z. , ’K £7/i<«16 QYZ+lKT+l30 T: ...
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