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MA232_Test1_solns - Test I S 6(a “‘0 m MA232 Fall...

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Unformatted text preview: Test I S 6 (a “‘0 m § MA232 Fall 2008 Differential Equations lYou must Show work to receive full creditl 1. Classify the following differential equations: (10 pts) (3) The Dym Equation (for solitons) : 52g _ 333u_0 6t 5323— ‘ 3"”araer (lePtvndch Vah‘ablc. "> bk lecpcndml" Variables ’2 75% (b) The Euler~Bernoulli Equation (for beam deflection) d4u _ L/‘Hd order depenJcm‘ vam‘ab/a -—3 Lg T O — 9 pf I‘m/(pendrrrf Vam‘ablé‘a x + o. S pf. 0 b5 Irweqr’ Vlow—rhamagenCOU) Show work? \— 2. Verify that y is (or is not) a solution to the differential equation and determine the value of C which satisfies the initial condition. (note: only 1 of the 3 is not a solution) (20pts) (a) y’=t -— y, y=Ce—t + t - 1, y(0)=10. , -25 = — c -15 t \I , 6 4’ 10:06.5”.6:;j7 0‘] '06 + i = fi— (6' ~/f H 36 1 ,f; @- (b) ty’ + 3y=2t5, y=§t4 + Ct—3, y(2)=1. \; ’> 13+ 6 mam s zH‘fiwig): 2235 £44Z£4 3g {Ll-P Sét'fi: 265 X . (c) ty’ — 3y=t3, y=t3 (0+ mm), y(1)=17. l \ \l‘;3t2[(+1n[£)]4 89;) 17 ~ ’(540) WWW/Mm C3? 1 x ‘ 3 55351610“)3+{Z§'3%*3[0M"EHglf I+ZPB'E MWE t3~ EQEW’QL‘J] 5-! 3 / m2» {.141 i > >1 fl 1. _ . IAIIIIIHJ 22222222221 4111'}? 15:11/222/I 22222222. 2 Txxxxxxxx|alzlzzxifiz It'll/Ix:~,///Ill..l __._____: __ \\\\\\\\\..III/////// . 11111112222/IIIIIII; r2222: 2; ‘\\\\\\\\\rm:///Ixzx//§ . m IIIII/Ix22/IIIIIII 2222222222 :xxxxxxx‘wIIII/x/2/ N m fiIIIIII//:://l IIIII ,.. 1/x/22/xz/xz/z/xxx/zf \\\\\\\\\|TI//////// e IIII////:..///Illlll //////////////////// :\\\\\\\\xMII//////22:. II. M 14:1111/222H1111H1 22/2//////:////T. ::\\\\\\T////222 u {nil/22: xtlnlm: 2222222222 .:.:::\\W:/2222.. W iii/122:1; :1 2.2222 _ :2::\\7/2222 e ttttt [Ix/~.//Illll __:______:_ :_ A __.___‘.\\7,..__.:_ M y::::://2_2//I:zc:a 2222222222 y 2222232222. . Lam IonlII/1222//IIIIII.. ////////222,///////.4 22:22’zlm\\\\::: d IIIIIII/2:,//III.»II. zzzzzzxxzzxxxxzzzxzz .22222,///L\\\\\~\‘\2$ . & IIIIIII/22/IIIIIII ////////////////////é 22/22/x/IIT\\\\\\:\ 7| d I:«:/://2://IIIII1~.. 2222222222 222///1r\\\\\::2 .Mw lvlllII/z..z//IIIIII .2..¢..2....2 /////////Im\\\\\\\\\\f K fi [I‘ll/2.2.,z/rouia 222222 2://///:_x\\\\\:: w n [1,512,271,]:i2‘ 22222: 2 2 , u ,4 O ////////Ill\\\\\\\\\ 0 .u sssss II/22111111a: 2222222 2 2 ///////II:_I\\\\\\\\\ II c III/I :1 11111 r hi, .7 d 2., ‘ f I \0 m ,1 Ix:_/// 4 . . w 2. ¢ r/xx/IIIIII_£\\\\\\\\\.4 rm \) LI m m.\ o n n O C e h t . m . . + .m (2 — 0! LI 5 1‘ w p + .m . Z P k a u .T IT . a w, la 0 + J .L Ila / .1 _C.|0 I. .d .1 j e \ m x g! \e w n AM If I I p! m /r f! a k In. a d: It 7. W. .T , IT M n rx n . W ITu M In. ‘1 e o h , 4w 0 7 a 2 .d ”T w h ,W :T k .19 P h ( ,7 In .. t 2 C 2 IT 47 IT 1T .w n h I. I ~ 8 ~ [ \l .1 C [T m) y r w 1 o“ z w d W S w mm = m u __ n 1 m __ A M 0 / 5 W I 0 5 .T w I \D M 3.0 y M + y y c y r + @M o a m 1 w m , O a a; B 1,, . v n c @ v m m. V 4. Solve the initial, value problems by separation of variables. (15 pts each) (a)2yy'=\/;J_—16, y(5)=2‘ a5 1-): :4 .— S/Vymu' ‘l-VVt'vlgHC k r-‘ .u={z.ng MM» C +( A c ( aw ‘ ’1 L/> V7546 ‘“ V’Tv—E’Zf‘; dfiztaq 2 7K } \/>:* Vila; -+( ow' M Iva fit 0 \JOJ‘hS / ' (b) y’=4t3y-y7 y(1)=-3 , {'4‘ 133—" WW3") 32‘”. 36 25 KM __ ng : “Vi-£0379 , .1 [WIT/z iq/f+6 /\ 1‘1 If C HI: 6 6 6 W": +~ c 6 MPH DC (c) y’=%;—§, y(—1)=0. ! ‘ .L. t w‘ {$34 ' £43 a? \I: 5+3 91: [9J‘Jzé‘l’ 6% m=4d éfil‘ é’dW-‘aq é‘l £9414) = ju/Wl‘L “Al's, C 1+” :6 ~ \H C C [47"5) 1" y: H 5 {H3} “NH «15(- a” H 'I N (\ 4! : (\ u I M /l \I: 7/: ‘I—‘g—‘Li fiqlfit-i—d /// ...
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