Test 2 spring 07 - NAME#1 Recitation Instructor_——— Signature Recitation Time Elementary Differential Equations Math 240 Spring 2007 Second

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Unformatted text preview: NAME #1 Recitation Instructor _____.____——— Signature __________.. Recitation Time Elementary Differential Equations Math 240, Spring 2007 Second Examination, March 13, 2007 Show all your work in the space under each question. Please write legibly and organize your solutions in a logical and coherent form; answers which are illegible or confusing will not receive credit. Each problem is worth 10 points. 1. Find the general solution. 2y” — 719' + 33/ = 0 307170 law @043 ivy—9:0 we 30:}. \l A’ (fldflk tail“ 2. Find the general solution to the system of equations. 9%Uid—V02g slfl '1 3+” {5% “7% iii, $ij - , Aim?” 7’1? 9% “31%. 10¢ x N7 an?» ‘%% “67x10 O\7’7\ _ a __ \ Kg; 753% \07k «9 DL’ 3b “we QQ—ZflKDALfl :b arc we: 98* Jr L16 , a?” — )C ,‘H' 1 6+ \) ’l’bfi 27" %C\€6 ,LLQ/ .- ’Z/C\& - ~"Ur x)? \figéfi ~3L~Vfl 3. Find the general solution. 3;” + 4y = 3x2 ¥f%uvo Evilk \%;<flu»1%xralaM2¢\ hfl 9&4w3x6»xc 93>2$v\%fi 8931“ LIX 4. UMMCL¥ {37x 4 a» 2378” LHHF*HB#%1A4QL23¢— quB ugco 7)\VALPLD a» (wax w Lamaze AC*% L‘% 4. A mass of 2 kg causes a spring to stretch 9.8m. This is attached to a damping mechanism with damping constant 5 The mass is allowed to come to equilibrium and is then set into motion by displacing it 3m and releasing it. Find the function a:(t) which gives the position of the mass at time t. ”‘ my fiAL ‘ ammo» \A<«$\ a 7” \4 158%70 iuio NOV?» who 71?” +emLz O i \3 Lo WA :0 are [L 5. Find the general solution. CD7” «1% WW \: 0 mm + AWN m b 1 1J—nglx «11 “LY - \)2L\@ 4 (do J< LgLosu JV “W14: 6. Find the general solution. 1+e (fab nab {0 +\\k\om:o D—xx my; \jh: (4 Qj\ AV Lia/17k W Ta} 1 2\%\1+a\— a“ firm w“ u 2 6* MgflflfM-Z’ gw*/L (W \AAA H—Ux ‘— 45* \0}5\’H03L\ “in m — \ogmhafi 2 a? \%\ HEW "éb 93“ WagVH 6 \\ (1r 43 a?» afief’k + (o"‘+efi°”‘\ MMHM 7. A mass of 1 kg is attached to a spring with spring constant 1 5:%. After coming to equilibrium, the mass is set into motion with a variable force of cost N, but is not given any initial displacement or velocity. There is no damping. Find the function :c(t) which gives the position of the mass attimet. K Pa $W¥ic-ont sumac $Mm20 5 . ‘ (Ll DLHCO A 0:)L‘k 74“: g Lost +LLQMY‘ 74g): Log’l *l’ Xfick%&%MPMfllWW%l®UMf N9“ 3 ’Pxérpl/ '* Premier l M l—wstl Jr 8 L054: Jr 1% cost + Rt L» $9608 : ~7/Jx QVWC J'le cost *le LBS/C "Rt mid 9U 7km 1 owe): 3% £19;th +50 gut O”; MMZM Wm) 7km? LLSVWE 4 at wt Whip-o: (“inset % item % Lfimgk Q ~— 70(0fl2w MALWM) 3 iltl" Khaki 8. A mass of 6199 is attached to a spring with constant 8 There is no damping. The mass is set into motion with an initial displacement of 2m and an initial velocity. It is observed that the amplitude of the resulting motion is 5m. What was the initial veolcity? 9. Consider the differential equation describing a mass-spring—damping system. 1 x" + 013' + Ex = F0 cos wt, 33(0) = 1, x'(0) = —— (a) Suppose F0 = 0. For which values of c is this critically damped? >4" >v L70 % A; $10 U‘KMW W nevi/w LLALtlpmio 9”»‘vtlhl’l‘ro :2) C»\ (b) If F0 = 1 and c = 0 What values of a; cause resonance? 74“ “37k: nois waigqu a ‘PLW MW w‘/%/‘ (c) If F0 = 0 and c = 0, the solution is m(t) = 008% + x/gsin (You may assume this is true.) Write x(t) in theaform m(t) flosfiut — ¢) for some w and qb. x _ g); 025%; W/” 9; (gaugiqu so Loaf/L l" ggmg : (LAM $0 U”\ \m’fia PM Ch QW‘ 1 “E s “EEK FEM“— 7’3 3; US} , A i”? 95 wsgtfiéwée \22/ Wist/k \ ’ Mi} 3\ 10. Match the graphs to the equations. Each graph ranges from 0 to 20 on the horizontal scale and from —10 to 10 on the vertical scale. (a) 2:” + 1033’ + 323 = 0 93(0) = 0, m’(0) = 100 b) as” + 16110 = 0 ar(0) = 0, x’(0) = 30 0) 5:5” + x’+10:c = 0 :r(0) = 0, x’(0) = 10 d) 1:” + 400:1: = 200 cos 18t 33(0) = 0, x’(0) = 0 e) x”+:r’+a:=cost 32(0) =0, 33’(0) =0 ( ( ( ( i I “\KXXTX‘KH‘E—h Nlrl rerl’VJl’lm'nlll I lia'lll This is the graph of equation (A . This is the graph of equation Cl . This is the graph of equation Q1 . This is the graph of equation This is the graph of equation C“ . ...
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This note was uploaded on 04/07/2008 for the course MATH 240 taught by Professor Yetter during the Spring '07 term at Kansas State University.

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Test 2 spring 07 - NAME#1 Recitation Instructor_——— Signature Recitation Time Elementary Differential Equations Math 240 Spring 2007 Second

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