AK-MATH225F10-EX1

AK-MATH225F10-EX1 - MATH 225 Fall 2010 Name: 30 OAS Exam I...

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Unformatted text preview: MATH 225 Fall 2010 Name: 30 OAS Exam I Section: In order to receive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all reasoning and work is provided. Where appropriate, please enclose your final answers in boxes. 1. (8 points) For the following, state the order of the differential equation and determine whether it is linear or nonlinear. t. _ y 2 3 MW” ‘‘‘‘‘ _, \ (a) %_6(%> =0 (Znalvovolcf/ Hmlc‘oear“ E 7"¥Cp (7’)3:O W (b) y”+w3y’+6y=e“ I'Zna/l“orvl€f"/ Decaf I awe—>4, a.tx)=><3, Camus gmwx (c —~=y—6 {H a» order, Liar I///_' I, =.. 7 Mazuzsvfmd ><>=<>« i:£§::.:2§:£32;22 W r (d) 1! (2+9): ’ I] -Gr‘oler/ Mon lirah‘] d 2. (12 points) Let 2% = 8y — 2y3, (a) Sketch the phase line (portrait) and classify all of the critical (equilibrium oints. ,LalH) E? ', Z7LH'7Z) :0 7:2, as 5:126 - _ Z": 9 M "" r Z)'0 L, 7 O r org/Fob) (1/) / y: _. 7:0, oosEAL/c/ (‘90’ repellcf; (o’o) 6’“ scarce, ((9/4) ' 1" "Z , as ,séab/e —%% " fl glvz :0 7 ora+FZa¢tor// or sink. (b) Next to your phase line, sketch the graph of solutions satisfying the initial conditions: y(0) = 0, 0 = —1, 1 = 1, 0 = 4 y() y() y() 666 Odom/gr l 3. (16 p01nts) Water containing Z kilograms of salt per liter IS pumped into a tank containing a brine mixture 9:33, gogt a rate of 80 liters per hour. The well-stirred mixture is pumped outat a rate of 100 liters per hour. I" (a) Let t denote the time in hours, and suppose that the volume of the mixture is 800 liters at time t = 0. Find V(t), the volume of the mixture at time t. Wt): <800 + t(?§o~_loo) / (c) Find the general solution of the differential equation from Put your solution in explicit form. Assuming t Z 0, What is the largest time interval on which the solution is valid? 3% + ( Hit) 2'20 ét—Jt 'SQo/LIO-tl a L; ~65 E3; - Aim-42" * :: a : a“ O Jostékm W'U'MJ ’ ‘ : (How‘s (was 4%: t 7: 20 (Hortfi; ( (HO,.t)'§7)’ 1“ 2,c>( LID—wt) ‘ word's} : Salt/O‘tygol‘? CLIO-tfg = +SUIOJCVH+C (d) If it is also known that 136 kilograms of salt are in the brine mixture at time t = 0, how much salt will be in the mixture 20 hours later? 7 Lo} : 51%) + CWOf #39 4. (18 points) Let m(t) and y(t) denote the amounts of two radioactive substances remaining at time t. The following system is a model of the radioactive decay series for the two substances. dm E dy B? Let a:(0) = 40 and y(0) = 0. Find a:(t) and y(t) using a method(s) of your choice. % “0&qu godf D “‘J‘ = —2w = —3y+2x Ax _ ' M __ “w ‘9‘ 'a-t ‘3? "Ex 47% '" “37+ u H05" J—Ax = 1%: xio/W 75 , z "2’6 5 x S ’ @217: 7% t 373 3:)" 34; cm : rum Mm == 6, i ._.»_ C C 3i: , -235 5 x :— + c352“? 6' +35%? : <30 C" a .' t - t Wee Lag; w v / . Cy”; : S YchJJC XCO) - file = H o r f ,WW at ’éét ,, 71—“ <50 6.“ “+5. X a H05 7 «at ~3‘t . g = 3 O C + Ca «Iota m (a? Fmtor (Oct/aw! misc worés L7 5. 16 poi ts) Solve the following difierential equations. . F‘X (a)4y”-4y’+y=0/ 7:6 :75 HFZWL’lF'I'l :0 Lanna—0:0 32% A}: P=*z’: /{ : 5:646: +"CLX@ (b)4y”—4yl+y=2e%+2 .21 7k: C’cl-J'CZXcZ 3w “hm/c, X. .X. dzz 7fzdeéZ+A/7fl:fo<)(¢ 4—erz 21-» i l Z+PT<L7><€Z N: Z0467, +0C)§(LZ #- 0< X6 1 . _ 2 His; wat fixwa’éé‘] vH [Maégméj + w +4 {fa Mnme :2 (4:3 A JL 2;: ‘ €04;_L / ~ \Z i 3 ;Céz/f.cz)wz,+xéa,fa 0< H 7 _ / LI Lt " N _____ "M mmmmmmmmmmmmm ,,,,,, v. 6. (14 points) A mass weighing 32 pounds stretches a spring 2 feet. Initially the mass is released from rest at a point 4 inches above equilibrium. (Use the convention that displacements measured below equilibrium position are positive.) (a) Find the equation of motion if the mass is driven by an external force of f (t) = 10 cos(3t). If applicable, put your solution in real form. (Your final solution should not contain complex numbers.) EE 4 M7l/4'K7ngt) 622’ Egg 7/, H69); jilocOs/x; 7w): ~ 13— / 7’(o):o 7k: 75U+th=O / 7hr‘Cr—t M7 :L5 Z Z I a tow =32 V “Hé‘O, P :Je/ {aim - j N;__ H515 _ tn K 32=uzl C “6 TCGSKH'CJ'F'cslrséR/t) ["9 7'“: C' Cosltft‘) +Ca57nlL/c} co: M ‘33 7F: 7f” + “9710: ’C‘COSB’C) 7’» " A'COS (BC) +5370 (31*) 7’; = '3451\n(_3t>+ BBcQSC'st) 7F" : — 914465136” fi Bsi‘o/sfl Vfidwslfil ‘Ol Belt-136'} mEkosZBtl +~ bgi‘nz’fidfl-flaeag fiflHo/fifl-él Wéio/ 5‘0 / 7F:“$Cos/3t> 7 m LLCoSZHt) + CZSFnLHt) + gags-t) : “WC, SzxnlLl‘C) '7‘" 1" Cacti—$191?) A‘Eé-E-Sl‘h, ( 3ft.) _ m _. l : ‘37 - I (O)‘Cl+-%D~ 3/ Cl 2,; / 71(0):L+CZ=o/CZ,*O (b) For which externa force f (t) = 10 cos(3t) or f(t) = 10 cos(4t) does the harmonic osciallator display resonance? 7. (16 points) Solve the given differential equation by means of a power series (centered at a: = 0). y/l+k2$2y = 0 where k is a constant. (a) Find the recurrence relation. ,, :0 Xo /; E ngxmn ,/ faém,_¢'>cw Xv?” 7* e / 7 - x - n13 V 0'] 0-”? .m r1»: 2 1 W X040 imam-flax + z x g5 n:& m N *1 Z- n+2.“ Z “CM/MM" +214 Cnx ~<r> r222. ":° N 0-2 .2 Z. = E admocnx" +2 K CH'HX O n:.7, 6:” ’3 'Z ' ' W n~2 9“ 2C DQKO +IC5XZ>C3X ~4— Z nirvacl, X + Z; KLC’PH X :0 h=H n: 9" ~z 1C3 + (9C3 x + Z Emmi) o, +L2C,,«,J X" =0 on: W 1%) WWW-me :0 mar-Q 2 __ zcz=o (acre '\ (Mich +z ore—o C :0 C 1C) : > Z . an OW /. ms . (b) Find the general solution up to degree 7, that is up to and including the 3:7 term (unless the series terminates sooner). (ya 7:” 2C0 x5 :Cé +'¢IX+CZ XZ+~ ~- - ~ 0:0 =~ o: c: _>_MM_WWWWWWWMWW.“ 1 — o n LC, )y-ce+clx .3. -__Z_ng\/__” “1 C; :C) a z R! Z s N co+c><~453X ~53” “:3343:0 Z 174— I [Z 20 . : “KZCO _ ' Z Co kWh-WW MMMMMMM 1.- ———————————————————————————— -. hzkl " CH H63) 12. neg ( C5 : ~——-.KZC:‘ : “LZC, (SM-0 7,0 v ’ — 45¢; ‘ m 'C , — _ CC (0X5) 0 ‘ w CD n17 ~ .—~ ' C7 (ma ...
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This note was uploaded on 02/08/2011 for the course MATH 225 taught by Professor Staff during the Fall '08 term at Mines.

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AK-MATH225F10-EX1 - MATH 225 Fall 2010 Name: 30 OAS Exam I...

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