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math242 Wint03 midterm1 and2

# math242 Wint03 midterm1 and2 - MATH 242 MIDTERM 1 W’03...

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Unformatted text preview: MATH 242 MIDTERM 1 W’03 Each problem is worth 10 points. Relax and good luck! NAME _- " PATIM v(_+) = (33% + (vi-r ‘37—‘5 \€ 1 { {3 MM) . ‘s \ V, “(1" H sat/<0 1. Show that the initial value problem ii— = 192‘”, y(0) = 0 does not have a unique ’} solution by exhibiting two distinct solutions. -3 7 ' a j _ f .. 3%W"L=© e ’ l X 1 ii ‘lz. ! ' _r dx XV 3X64 ><z\// eds/(0m- 3% W V1 1, mix, t J“)! . if i d -» 211W -— ' >31? >< :Cﬂgﬁ—i: 7‘ xi; . X W" /i_ a” i VCX) #C, 2 y in] z L ﬁC:O I: 20— 3(0) ZJ—ﬂ-PC y 3 ii T. W. W 1* _ All ‘ d dx W 5 Fiat W”? “A 37 0‘7“ 3X X 1W :— ,X:+C A \/(O):O X 205) — Q; :C as Q=O c : we 2 y =(hx7—+tc)”e “0 V :(‘q W1 #3 1y? “i it, :L 3 0i)< Hex % t 9.011 2 x35 'Fuwdaﬂuwﬂﬁi goMﬁb6V7l€+ ~ VWKWMWW What is lim y(x) , where y(x) is the solution to the initial value problem ya”): ~7yx4wlO><+C 52(0):} ‘4 I: yl'(o')-~‘7>//o)+10(05+C C5] '7 ﬁx): YV'1*7\/%+'Ox+‘ 1‘ (y 2—' 7+ l 0‘5 Y iyx 10¥+) 3 X_# f “l f N 5Yzﬁhodypgdy MKVE¢VH03:Y*C C in [( y- S‘a( 36%)] C. :eer (y—syy~z>:C€X ('~ (M A X 3. Consider the Bernoulli equation d—y—Q = —x2y2 . (1) dx x (a) Show that the substitution v = y“I reduces equation (1) to the equation ﬂ+2-‘i=x2. (2) dx x (b) Solve equation (2) for v. Then make the substitution v = y“ to obtain the solution to equation (1). d '2 l l—n -— —\ 0T)?“ xf'wz' Viv evty‘zevw OIV -2d :5 __ - ,_ 2-0;! — 3'..- —n 7— SUb dv 2 _ 'L dx >< “ y A “"‘ ‘" “‘1 (AV 6 7 dx X(v) ll Zth 7. 6570” :e "-x dx x d 1 __L (XZ‘Vi X —> J a éXS-i-C __) V: +- .X": Z X M N 3M _2 am _ __ Yr; 1 :- "-— f (1) {—Jdd3y—izjdyzo 37 8x y y M N r 3 W(x+3x3siny)dx+(x‘cosy)dy=o ~35! COS\/ 3X [1y CO )1 M M 1 3M x x Eli, .Y (iii) (1+exy+xe‘y)dx+(xe’+2)dy=0 79‘": 6 + X9 3% »- ‘Xe 4* £1 (Hexg +xex5‘3dxr (yeX r2)dtj :O M—FN 5. Suppose that a skydiver topples out of an airplane. The relevant initial value problem is = g ~(ﬂjv , v(0) = O , where g is the gravitational constant and b is the m coefﬁcient of air resistance. Assume that g =10 % . 1 (a) Suppose that 9— : ~5—. What is the terminal (limiting) velocity for the free-falling m skydiver ? (b) When does the skydiver reach 90% of the terminal velocity in part (a) ? . V(O\’O ) C5: '0‘ when VK—Hf—O ﬂ : go «— Soe‘VSt _\ 41+ “50 ’— SOe“‘—s+ A l :9 75+ a m1:an 5 ON ' b CW is" = —) 9.7L! 4» 3v- : ‘ d7: -— ‘3 a? S \1 O on; i Marv-:65“ :_ C 5 +511! 4, Cit/g i 6144500 9 0H) ‘35—“ V > Us ' “S i“ — 8095+ {\l—JEOl (53 V) :6 (103 ’9 G \l— - a pqxso : a; 6 Mmam 1 SOLUTWJS‘ l. One “WM. .2 10030. IT 710 J we J‘ﬂbwa‘rm 01C var-mug“: 41 = SW =7 11‘“: {N . 11 w“ )6 ﬂ x ‘1Cﬁ\:0=3 (-10% 2x1 —-—~ :3 \1_._‘_Z Z. 7": 71-17440 :(‘1—2.)(‘1~I) [Aumwamoux] Fl'xﬁJQ As'agram rkue )th W MATH 242 MIDTERM 2 W’03 Relax and good luck! NAME ’ MO” LP? ~53 Phi! m“ h D 7 " 10(1er g.) +(‘ 1. (40 points) The following differential equations model an idealized mass-spring system. For each equation, 2:3. ‘1 (i) Find the position of the mass at any time t. 7- 0‘ b ’3, (ii) Is the motion oscillatory? ‘7. 1 (iii) Is the motion periodic? Ifso, what is the period? 7- 1 (iv) Does the mass ever cross the mgﬁon? If so, when does it cross the equilibrium position for the‘firSt time? ’1. ' \ (v) Sketch the graph of the position function. 1 . I'L- (a) 3y”+48y ‘23-,y'(0)=2 (b) y'+5y'+4y=0,y(0)=1,y'(°)=1 a. 3r“ 148:0 em=3 , we, lung r: : +_ Lh- ‘ v E —— C- —‘r -—3 C\:'- 2- le+)=C1COSl++'+-C2Slnl‘i+ ﬂkdfoyifci ‘ r: a CL: 1/2— ija : ’L'ICISK‘WLH' + H CLCOS ‘iJr—b 9 (0)“ tic?"— ®'9-E+S;"—LCOSH++—ﬂlzﬁolnq‘l—2 ® 565 ‘Hne mo‘h‘on is oscillatory C 4/1 t — _.,.._'_. "'— —- 'z‘ 4.: "-3 gaAsin(wt+¢) ﬂame {a 'ilancb‘cz 'e | (b w ﬁsinCHt-T—D .szrﬂZTl’_[_31:"-, PM“? " T Hall l4€9l+t5p€m6dfc ‘ n gag. (Jr (VESqu Hoe Graeiilibrium pOS‘ho ' ® [50‘310 I do“? {.3 d 1 ST ? einCHt‘E):O ﬁH*‘%:ﬁ(S€*n“)q “.0 2. (16 points) Determine the form of a particular solution for the following [1. differential equations. 7; a (a) y’—8y’+25y=5x3e‘x—7e" ,2. :4 £2: 1 (b) y" +4y = 64" D (C) y'+y==x+sinx 0 (d) y'+2y'+2y=3e"‘+4cosx @ sz8r+ZS:O—’ rzl-ltai A 9(x)=e”"(cosa><+€m3><) A wx ' . I (3 We‘ﬁw‘“ —» gasp—qt)wwmxaexucmo) or” («0 iitgp=€“*(A»<3+Bx1+cX+D)] Lu. .7 ,7, ,..—__,,,_. ' Z (a r1+Hro a rli’Zu' rx (aha: Coslx‘rg‘r‘ X 9-2.x “>3 Jljpz Ae—ix‘} l ‘n 6) r1“ : O (a r: i‘ Lj(><): cosx +S\ X " S‘n‘x Y+an><~ak5plipxx+8 , tjpercoska \ so [/JF;AX+B+CCOS‘K*DS‘AK ‘ HEMP" XCAX*B*C¢OSX¥ ,,J - ,I_—>‘ 5y+§lﬁ><) z 2 2-04rsflti —> 3&ij (C0 cl ‘r + r* “I 3. (24 points) -- Find—the general solution to the differential equation -y"-—’4j74-4y=(x+1)ezx. ' r2““‘l|’+"f:0 ar:‘2_ v'ﬁrepeaﬁ’d 1‘1 LjCK): Clezwk-i’szezx l. (40 points) ' The following diﬁ‘erential equations model an idealized mass-spring system. For each equation, ‘ ‘ (i) Find the position of the mass at any time t. (ii) Is the motion oscillatmy? (iii) Is the motion periodic? If so, what is the period? (iv) Does the mass ever cross the equilibrium position? If so, when does it cross the equilibrium position for the ﬁrst time? (v) Sketch the graph of the position funotion. (a) 3y"+4sy= o ,y(0)= «yz J10): 2 (b) y'+5y'+4y‘=0,y(o)=1,y'(0)=1 (a) (1) l1: ~31 mLH: +-\2-_hn'-H' (m ‘(Ls EEC) Yes; but T=L%‘=J\$: 1;: 8") fLm-amwm few 4 m. “new; 7. {émkut-w/q) ﬂu: Cir-069:; when t: +:tK-T i h: UIIJZ-i‘“ ’ SN‘M'A “TWA Win“? t "5 (pr/K. u”) a} ‘1: gei'zs‘W-‘w (it) No (Lu) Mr, M Sam; §¢¢v—’§e'“f:o J m it..." zigﬂnﬁﬂo . so in mm mm With-«um» m 7 (0.41 ...
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