EXAM-3 -B2 MATH 261: Elementary Differential Equations MATH 261 FALL 2012 EXAMINATION COVER PAGE Professor Moseley

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1 EXAM-3 -B2 MATH 261: Elemetry Differetil Equtios MATH 261 FALL 2012 EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Lst Nme, First Nme MI (Wht you wish to be clled) ID # EXAM DATE Fridy, Oct. 29, :30m I swer d/or ffirm tht ll of the work preseted o this exm is my ow d tht I hve either give or received y help durig the exm. Dte Sigture SIGNATURE DATE INSTRUCTIONS: Besides this cover pge, there re 12 pges of questios d problems o this exm. MAKE SURE YOU HAVE ALL THE PAGES. If pge is missig, you will receive grde of zero for tht pge. Red through the etire exm. If you cot red ythig, rise your hd d I will come to you. Plce your I.D. o your desk durig the exm. Your I.D., this exm, d stright edge re ll tht you my hve o your desk durig the exm. NO CALCULATORS! NO SCRATCH PAPER! Use the bck of the exm sheets if ecessry. You my remove the stple if you wish. Prit your me o ll sheets. Pges 1-12 re Filli-the Blk/Multiple Choice or True/Flse. Expect o prt credit o these pges. For ech Fill-i-the Blk/Multiple Choice questio write your swer i the blk provided. Next fid your swer from the list give d write the correspodig letter or letters for your swer i the blk provided. The circle this letter or letters. There re o free respose pges. However, to isure credit, you should expli your solutios fully d crefully. Your etire solutio my be grded, ot just your fil swer. SHOW YOUR WORK! Every thought you hve should be expressed i your best mthemtics o this pper. Prtil credit will be give s deemed pproprite. Proofred your solutios d check your computtios s time llows. GOOD LUCK!! REQUEST FOR REGRADE Plese regrd the followig problems for the resos I hve idicted: (e.g., I do ot uderstd wht I did wrog o pge.) Scores pge poits score (Regrdes should be requested withi week of the dte the exm is retured. Attch dditiol sheets s ecessry to expli your resos.) I swer d/or ffirm tht upo the retur of this exm I hve writte othig o this exm except o this REGRADE FORM. (Writig or chgig ythig is cosidered to be chetig.)

2 Totl 100 MATH 261 EXAM 3-B2 Professor Moseley Pge 1 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. We do ot solve the differetil equtio L[y] = g(x) where L[y] = y + y is lier opertor tht mps A (R,R) to A (R,R) by isoltig the ukow fuctio. We use the lier theory. The dimesio of the ull spce of L[y] is 2. Sice the opertor L[y] = y + y hs costt coefficiets, we ssume solutio of the homogeeous equtio L[y] = 0 of the form y = e rx. This leds to the two lierly idepedet solutios y 1 = cos(x) d y 2 = si(x) so tht bsis of the ullspce of L is B = {cos(x), si(x)}. Hece we c deduce tht y c = c 1 cos(x) + c 2 si(x) is the geerl solutio of the homogeeous equtio y + y = 0. To use the lier theory to obti the geerl solutio of the ohomogeeous equtio L[y] = g(x), we eed prticulr solutio, y p, to y + y = g(x). We hve studied two techiques for this purpose (ttedce is required): i) Udetermied Coefficiets (lso clled judicious guessig) ii) Vritio of Prmeters (lso clled vritio of costts) For ech of the fuctios g(x) give below, circle the correct swer tht describes which of these techiques c be used to fid y p for the ohomogeeous equtio y + y = g(x): 1. (2 pts.) g(x) = 2x 1 e x. A B C D E 2. (2 pts.) g(x) = 3 e x. A B C D E 3. (2 pts.) g(x) = 4 sec (x). A B C D E A) Neither techique works to fid y p. B) Oly Udetermied Coefficiets works to fid y p. C) Oly Vritio of Prmeters works to fid y p. D) Either techique works to fid y p. E) Not eough iformtio is give. AB) Too much iformtio is give. AC) All of the bove sttemets re true. AD) Noe of the bove sttemets re true.

3 Totl poits this pge = 6. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM-3-B2 Professor Moseley Pge 2 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. Let y" 4y +4 y = x 1 e 2x I = (0, ) (i.e. x>0) be (*), L[y] = y" 4y + 4y, d N L be the ull spce of L. Begi the solutio of (*) d the swer the questios below. 4. (3 pts.) The geerl solutio of y" 4y+4y = 0 is y c (x) =. A B C D E 5. (4 pts.) To fid prticulr solutio to y" 4y + 4y = x 1 e x we let y p (x) = u 1 (x)e x + u 2 (x)xe x. Substitutig ito (*) d mkig the pproprite ssumptio we obti the two equtios:. A B C D E Possible swers this pge A)c 1 cos(x) + c 2 si(x) B)c 1 cos(2x) + c 2 si(2x), C)c 1 cos(3x)+c 2 si(3x) D)c 1 cos(4x) + c 2 si(4x) E)c 1 e x + c 2 xe x AB)c 1 e 2x + c 2 xe 2x AC)c 1 e 3x + c 2 xe 3x AD)c 1 e 4x + c 2 xe 4x AE) r = ±2i BC) r =±2i BD) 1,1 BE) 1,1 CD) u 1 (x) e x + u 2 (x) xe x = 0, u 1 (x) e x + u 2 (x) (e x +xe x ) = x 1 e x CE) u 1 (x) e 2x + u 2 (x) xe 2x = 0, 2 u 1 (x) e 2x + u 2 (x) (2xe 2x +e 2x ) = x 1 e 2x DE) u 1 (x) e 3x + u 2 (x) xe 3x = 0 3 u 1 (x) e 3x + u 2 (x) (3xe 3x +e 3x ) = x 1 e 3x ABC) u 1 (x) e 4 + u 2 (x) xe 4x = 0, 4 u 1 (x) e 4x + u 2 (x) (4xe x +e 4x ) = x 1 e4 x ABD) u 1 (x) e x + u 2 (x) xe x = 0, u 1 (x) e x + u 2 (x) (xe x +e x ) = x 1 e x ABE) u 1 (x) e x + u 2 (x) xe x = 0, 2 u 1 (x) e 2x + u 2 (x) (2xe 2x + e 2x ) = x 1 e 2x ACD) u 1 (x) e 2x + u 2 (x) xe 2x = 0, 3 u 1 (x) e 3x + u 2 (x) (3xe 3x +e 3x ) = x 1 e 3x ACE) u 1 (x) e 3x + u 2 (x) xe 3x = 0, 4u 1 (x) e 4x + u 2 (x) (4xe 4x +e 4x ) = x 1 e 4x ADE) u 1 (x) e 4x + u 2 (x) e 4x = 0, u 1 (x) e x u 2 (x) e x = x 1 e x BCD) u 1 (x) e x + u 2 (x) e x = 0, u 1 (x) e x u 2 (x) e x = x 1 e x

4 BCE) u 1 (x) e x + u 2 (x) xe x = 0, u 1 (x) e x u 2 (x) e x = x 1 e x BDE) u 1 (x) e x + u 2 (x) xe x = 0, u 1 (x) e x u 2 (x) e x = x 1 e x ABCDE) Noe of the bove. Totl poits this pge = 7. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Professor Moseley Pge 3 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. Let (*) be the the ODE L[y] = g(x) I = (0,) where L[y] is give secod order lier differetil opertor d ga ((0,),R) d let N L be the ull spce of L. Suppose tht i solvig (*) you hve foud the geerl solutio of L[y] = 0 to be y c (x) = c 1 e 2x + c 2 xe 2x. The followig the stdrd procedure to fid prticulr solutio of (*) usig vritio of prmeters, you let y p (x) = u 1 (x)e 2x + u 2 (x) xe 2x d tht this results i the followig equtios i u 1 (x) d u 2 (x) (see the previous pge for why this might be true): u 1 (x) e 2x + u 2 (x) xe 2x = 0 u 1 (x) e 2x + u 2 (x) (2xe 2x + e 2x ) = 2 x 1 e 2x Give this iformtio, you re to fid u 1 (x) d u 2 (x) d the fiish the solutio of (*) o the ext pge. If your u 1 (x) or u 2 (x) is wrog, the your solutio o the ext pge will be wrog. 6. (2 pts.) Hece u 1 (x) =. A B C D E 7. (2 pts.) Ad u 2 (x) =. A B C D E 8. (3 pts.) We my choose u 1 (x) =. A B C D E 9. (3 pts.) Ad u 2 (x) =. A B C D E

5 Possible swers this pge. A) 1 B) 2 C) 3 D) 4 E) 1 AB) 2 AC) 3 AD) 4 AE) x BC) 2x BD) 3x BE) 4x CD) x CE) 2x DE) 3x ABC) 4x ABD) x 1 ABE) 2x 1 BCD) 3x 1 BCE) 4x 1 BDE) l x CDE) 2 l x ABCD) 3 l x ABCE) 4 l x ABCDE) Noe of the bove Totl poits this pge = 10. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Professor Moseley Pge 4 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios Let (*), L[y], d N L be s o the previous pge. Usig the dt from the previous pge, fid prticulr solutio d the geerl solutio to (*). 10 (2 pts.) A prticulr solutio to (*) is y p (x) =. A B C D E (Recll tht prticulr solutios re ot uique. Use the procedure give i clss. (Attedce is mdtory.) 11 (2 pts.) The geerl solutio to (*) my be writte s y(x) =. A B C D E 12. (2 pts.) A bsis for N L is B =. A B C D E (Recll tht bsis is ot uique. Use the procedure give i clss. (Attedce is mdtory.) Possible swers this pge. A) x 1 B) x 1 C) x 2 D) x 2 E) l x AB) l x AC) x l x AD) x l x AE) (x+ x l x) e x BC) 2 (x+ x l x) e 2x BD) 3 (x+ x l x) e 3x BE) 4 (x+ x l x) e 4x CD)(1+ l x )xe x CE) 2(1l x )xe x DE) 3(1+ l x )xe x ABC)4(1 l x )xe x ABD)c 1 e x + c 2 xe x + (x+ l x) e x ABE) c 1 e 2x + c 2 xe 2x + 2(x+ x l x) e 2x

6 ACD)c 1 e 3x + c 2 xe 3x + 3(x+ x l x) e 3x ABC)c 1 e 4x + c 2 xe 4x + 4(x+ x l x) e 4x ABD)c 1 e x + c 2 xe x +( 1 + l x )xe x ABE)c 1 e x + c 2 xe x + (1 l x )xe x ACD)c 1 e x + c 2 xe x + ( 1 + l x )xe x ACE)c 1 e x + c 2 xe x + ( 1 l x )xe x ADE){1, x} BCE){1, x 1 } BDE) {x, x 1 } CDE){e x, xe x } ABCD){e 2x, xe 2x } ABCE){e 3x, xe 3x } ABDE) {e 4x, xe 4x } ABCDE) Noe of the bove Totl poits this pge = 6. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Professor Moseley Pge 5 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. The dimesio of the ull spce of the lier opertor L[y] = y y tht mps A (R,R) to A (R,R) is 2. Assumig solutio of the homogeeous equtio L[y] = 0 of the form y = e rx leds to the two lierly idepedet solutios y 1 = 1 d y 2 = e x so tht bsis of the ull spce of L is B = {1,e x }. Hece we c deduce tht y c = c 1 + c 2 e x is the geerl solutio of the homogeeous equtio y y = 0. Use the method of udetermied coefficiets s discussed i clss (ttedce is mdtory) to determie the proper (most efficiet) form of the judicious guess for prticulr solutio y p of the followig ode's. Choose the correct (most efficiet) fil form of the judicious guess for prticulr solutio y p of the followig ODE s. 13 (3 pts.) y y = 2 si x First guess: y p = Secod guess (if eeded): y p = Third guess (if eeded): y p = Fil guess. A B C D E 14. (3 pts.) y y = 3 e x First guess: y p = Secod guess (if eeded): y p = Third guess (if eeded): y p = Fil guess. A B C D E 15. (3 pts.) y y = 4 xe x First guess: y p = Secod guess (if eeded): y p = Third guess (if eeded): y p = Fil guess. A B C D E Possible Aswers for Fil Guesses. A) A B) Ax + B C) Ax 2 + Bx + C D) Ax 2 E) Ax 2 + Bx AB) Ae x AC) Axe x AD) Ax 2 e x AE) Axe x + Be x BC) Ax 2 e x + Bxe x BD) Ae x BE) Axe x

7 CD) Ax 2 e x CE) Axe x + Be x DE) Ax 2 e x + Bxe x ABC) A si x ABD) A cos x ABE) A x si x ACD) A x cos x ACE) A si x + B cos x ADE) Ax si x + Bx cos x BCD) A x si x + B x cos x + C si x + D cos x BCE) Udetermied Coefficiets works o this problem, but oe of the bove is the correct form. BDE) Udermied coefficiets does ot work for this problem. Totl poits this pge = 9. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Prof. Moseley Pge 6 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. Also, circle your swer. Let y IV 4y= 0 be (*). Solve (*) below or o the bck of the previous sheet. Also let L:A(R,R) A(R,R) be defied by L[y] = y IV 4y. Be creful s oce you mke mistke, the rest is wrog. 21. (1 pt). The dimesio of the ull spce of L is. A B C D E A) 1 B) 2 C) 3 D) 4 E) 5 AB) 6 AC) 7 ABCDE) Noe of the bove. 22. (2 pts). The uxiliry equtio for (*) is. A B C D E A) r 4 r 2 = 0 B) r 4 4r 2 = 0 C) r 4 9r 2 = 0 D) r 4 16r 2 = 0 E) r 4 4r 3 + 4r 2 = 0 ABCDE) Noe of the bove. 23. (3 pts). Listig repeted roots, the roots of the uxiliry equtio re. A B C D E A) r = 0,0,1,1 B) r = 0, 0, 2, 2 C) r = 0,0,3,3 D) r = 0,0,i, i E) r = 0, 0, 2i,2i AB) r = 0, 0,3i, 3i AC) r = 0, 0, 4i, 4i ABCDE) Noe of the bove. 24. (3 pts). A bsis for the ull spce of L is B =. A B C D E A)){1, x, e x, xe x } B) ){1, x, e 2x, xe 2x } C){1, x, e 3x, xe 3x } D)){1, x, e 4x, e 4x } E){1, x, e x, e x } {1, x, e 2x, e 2x } AC) {1, x, e 3x, e 3x } AD){1, x, e 4x, e 4x } ABCDE) Noe of the bove. AB) 25. (3 pt). The geerl solutio of (*) is y(x) =. A B C D E A)c 1 +c 2 x + c 3 e x + c 4 xe x B)c 1 +c 2 x + c 3 e 2x + c 4 xe 2x C) c 1 +c 2 x + c 3 e 3x + c 4 xe 3x D)c 1 +c 2 x + c 3 e 4x + c 4 xe 4x E) c 1 +c 2 x +c 3 e x +c 4 e x AB) c 1 +c 2 x + c 3 e 2x + c 4 e 2x AC) c 1 +c 2 x + c 3 e 3x + c 4 e 3x AD) c 1 +c 2 x + c 3 e 4x + c 4 e 4x ABCDE) Noe of the bove.

8 Poits this pge = 12. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Prof. Moseley Pge 7 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios y + 2y = 6e x + 16 be (*). Solve (*) o the bck of the previous sheet. 21. (3 pts.) The geerl solutio of y + 2y = 0 is Let y c (x) =. A B C D E 22. (5 pts.) A prticulr solutio of y + 2y = 6e x is y p1 (x) =. A B C D E 23. (5 pts.) A prticulr solutio of y + 2y = 16 is y p2 (x) =. A B C D E 24. (1 pts.) A prticulr solutio of (*) is y p (x) =. A B C D E 25. (2 pts.) The geerl solutio of (*) is y(x) =. A B C D E Possible swers this pge A) c 1 + c 2 x + c 3 e x B) c 1 + c 2 x + c 3 e 2x C)c 1 + c 2 x + c 3 e 3x D) C)c 1 + c 2 x + c 3 e 4x E) c 1 + c 2 si(x) + c 3 cos(x) AB) c 1 e x + c 2 si(x) + c 3 cos(x) AC) ) e x AD) 2e x AE) ) 3e x BC) 4e x BD) x 2 BE) 2x 2 CD) 3x 2 CE) 4x 2 DE) 6x 2 ABC) 8x 2 ABD) e x +2x 2 ABE) 2e x + 4x 2 ACD) 3e x + 6x 2 ACE)4e x +8x 2 ADE) 2 e x + 2 si(x)

9 BCD) 2x 2si(x) BCE) e x +2x 2 + c 1 + c 2 x + c 3 e x BDE)2e x +4x 2 +c 1 +c 2 x+c 3 e 2x CDE) 3e x +6x 2 +c 1 +c 2 x+c 3 e 3x ABCD)4e x +8x 2 + c 1 + c 2 x + c 3 e 4x ABCE) 2x + e x +c 1 si(x)+ c 2 cos(x) + c 3 e x ABDE)2e x +2si(x)+cos(x)+c 1 e x + c 2 e x +c 3 x ACDE) 2e x +2si(x)+ cos(x)+c 1 xe x +c 2 e x +c 3 xe x BCDE) 2e x + 2 si(x) + c 1 e x + c 2 xe x + c 3 ABCDE) Noe of the bove. Totl poits this pge = 16. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Prof. Moseley Pge 8 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. O the bck of the previous sheet fid recursio formul for fidig the coefficiets to power series solutio bout x = 0 to the ODE y + x y 2y = 0 which we cll (*) 26. (1 pts.) To fid the recursio formul for the power series solutio of (*) bout x = 0 we let y =. A B C D E 27. (2 pts) Substitutig this series ito (*) (d o further computtios) we obti the equtio. A B C D E Possible swers this pge A) x N B) x C) D) E) 1 x ( 1)x AB) x 2 1 AC) ( 1)x x x x AD) 0 ( 1)x x 0 x 2 0 x AE) 0 ( 1)x x 0 x 3 0 x BC) 0 ( 1)x x 0 x 4 0 x BD) 0 ( 1)x 0 x 0 x x

10 2 1 BE) CD) ( 1)x x x 2 x 0 ( 1)x x 3 x 0 CE) 2 1 ( 1)x x x 4 x 0 DE) ( 1)x 2 0 x 0 ABCDE) Noe of the bove. Totl poits this pge = 3. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Prof. Moseley Pge 9 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. Let (*) be s o the previous pge. 28. (3 pts) As explied i clss (ttedce is mdtory) by chgig the idex d simplifyig, 2 the term ( 1)x c be chged to obti 0 0 ( 1)x 2 =. A B C D E 29. (3 pts) Cotiuig the procedure give i clss, usig this ew term d other simplifictios, the equtio you obtied o the previous pge c ow be writte s. A B C D E 30. (3 pts.) The recursio formul for fidig the coefficiets i the power series solutio of (*) is. A B C D E Possible swers this pge. A) ( 2)( 1)x B) ( 2)x C) D) ( 2)( 1)x E) AB) ( 2)( 1)x [( 2)( 1) ( 1) ]x 0 AC) 2 AD) [ ( 2)( 1) ( 3) ]x 0 AE) ( 1)x 2 [( 1)( 2) 0 2 ( 2) ]x 0 [ 0 2( 2)( 1) ( 4) ]x 0

11 1 1 BC) [ BD) BE) 0 1( 2)( 1) ( 2) ]x ( 1) ( 2)( 1) CD) 2 CE) 2 DE) 1 ( 2)( 1) ( 2)( 1) ( 2)( 1) ABCDE) Noe of the bove Totl poits this pge = 9. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Prof. Moseley Pge 10 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. Let y + p(x) y + q(x) y = 0 be (*). Suppose (*) hs vrible coefficiets d tht the solutio of (*) by power 1 series leds to the recursio reltio 2 For = 0, 1, 2, 3, 4,... As illustrted i clss 1 (ttedce is mdtory), you re to fid the (first four ozero terms i the) power series solutio to the iitil vlue problem. IVP ODE y + p(x) y + q(x) y = 0 IC s y(0) = 2, y(0) = (1 pts.) 0 =. A B C D E 32.(1 pts.) 1 =. A B C D E 33. (4 pts.) The power series solutio to this IVP is y(x) =. A B C D E Possible swers this pge.

12 A) 0 B) 1 C) 2 D) 3 E) 4 AB) AC) x x x AD) 3 3x x x AE) BC) x x x BD) 1 x x x BE) AD) x x x ABCDE) Noe of the bove x x x x x x x x x Totl poits this pge = 6. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Prof. Moseley Pge 11 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. MATHEMATICAL MODELING. As doe i clss (ttedce is mdtory), o the bck of the previous sheet, you re to develop geerl mthemticl model for the mss/sprig problem. Tke positive distce to be dow. Suppose poit prticle with mss m due to its weight W = mg where g is the ccelertio due to grvity stretches sprig of legth L distce Δ. If the mss is stretched dowwrd distce u 0 from its equilibrium positio d give iitil velocity v 0 i the dowwrd directio, develop pproprite mthemticl model to determie the subsequet motio (i.e. to fid the distce u(t) from the equilibrium positio s fuctio of time). Assume tht the ir resistce is lier with proportiolity costt c > 0 i feet slugs per secod d tht the sprig costt is k i pouds per foot (or slugs per secod squred). Assume exterl force g(t) i slug feet per secod squred. 34. (2 pt) The fudmetl physicl lw used to develop the ODE i the model is. A B C D E A) Ohm's lw, B) Coservtio of mss C) Coservtio of eergy D) Newto's secod lw (Coservtio of mometum) E) Kirchoff's voltge lw AB) Kirchoff's curret lw (Coservtio of chrge) ABCDE) Noe of the bove. 35. (2 pt.) Usig sttics, we my obti the reltioship betwee Δ, k, m, d g s. A B C D E A) k = Δ m g B) k Δ = m g C) k m = Δ g D) k g = m Δ E) k m Δ = g ABCDE) Noe of the bove. 36. (4 pts.)the geerl mthemticl model tht describes the dymics of the mss sprig system whose solutio yields the distce u(t) from the equilibrium positio s fuctio of time is. A B C D E A) mu cu ku g(t) B) mu cu ku 0 C) m u k u g ( t ) D) m u k u g ( t ) E) mu cu ku g(t), u(0) u0 u(0) v0 AB) mu cu ku 0, u(0) u 0 u(0) v 0 AC) mu ku g(t), u(0) u0 u(0) v0 AD) mu ku 0, u(0) u 0 u(0) v 0 ABCDE)Noe of the bove. 37. (1 pt.) The uits for the ODE i this model re. A B C D E A) Feet, B) Secods C) feet per secod, D) feet per secod squred, E) Pouds, AB) Slugs, AC) Slug feet AD) Noe of the bove.

13 Totl poits this pge = 9. TOTAL POINTS EARNED THIS PAGE MATH 261 EXAM 3-B2 Prof. Moseley Pge 12 PRINT NAME ( ) ID No. Follow the istructios o the Exm Cover Sheet for Fill-i-the Blk/Multiple Choice questios. Also, circle your swer. MATHEMATICAL MODELING. Cosider the followig problem (DO NOT SOLVE!): A mss weighig 4 lbs. stretches sprig (which is 12 ft. log) 3 iches. If the mss is lowered 4 iches below its equilibrium positio d give iitil velocity of 4 ft./sec. upwrd, determie the subsequet motio (i.e. fid the distce from the equilibrium positio s fuctio of time). Assume tht the ir resistce is egligible d tht there is o exterl force. O the bck of the previous sheet, pply the dt give bove to the model you developed o the previous pge to obti the specific model for this problem. DO NOT SOLVE! The swer the questios below. 38. (3 pts.) The sprig costt k i pouds per foot (or slugs per secod squred) is k =. A B C D E 39. (2 pts.) The ODE i the specific mthemticl model for the mss sprig system from the dt bove whose solutio yields the distce u(t) dow from the equilibrium positio s fuctio of time is. A B C D E 40. (1 pt.) The iitil positio for the specific mthemticl model for the mss sprig system from the dt bove whose solutio yields the distce u(t) dow from the equilibrium positio s fuctio of time is u(0) =. A B C D E 41. (1 pt.) The iitil velocity for the specific mthemticl model for the mss sprig system from the dt bove whose solutio yields the distce u(t) dow from the equilibrium positio s fuctio of time is u(0) =. A B C D E Possible swrs this pge.

14 A)1 B)2 C)4 D)5 E) 6 AB)4/3 AC) 8/5 AD)32 AE)48 BC)3 BD) 4 BE) 5 CD) 6 1 u u 0 16 CE) 1/6 DE)1/3 ABC)1/4 ABD) 1/2 ABE) 5/(12) ACD) 3/2 ADE) BCD) u 3u 0 8 BDE) u u 0 CDE) u u 0 ABCD) ABCE) ABDE) ABCDE) Noe of the bove. Totl poits this pge = 7. TOTAL POINTS EARNED THIS PAGE 1 u 4 u u 48u u 12u 0 8

EXAM-3 -A4 MATH 261: Elementary Differential Equations MATH 261 FALL 2010 EXAMINATION COVER PAGE Professor Moseley

EXAM-3 -A4 MATH 261: Elementary Differential Equations MATH 261 FALL 2010 EXAMINATION COVER PAGE Professor Moseley EXAM-3 -A4 MATH 261: Elemetry Differetil Equtios MATH 261 FALL 2010 EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Lst Nme, First Nme MI (Wht you wish to be clled) ID # EXAM DATE Fridy, Oct. 29,