s250ex2_fa07_

# s250ex2_fa07_ - Math 250 NAME Fall 2007 Exam 2 ID No...

This preview shows pages 1–10. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 250 NAME: Fall 2007 Exam 2 ID No: SECTION: This exam contains 10 questions on 10 pages (including this title page). This exam is worth a total of 100 points. The exam is broken into two parts. There are six multiple choice questions, each worth 5 points, and 4 partial credit problems. To receive full credit for a partial credit problem all work must be shown. When in doubt, ﬁll in the details. No notes, books or calculators may be used during the exam. Please, Box Your Final Answer (when possible). Multiple Choice Section 1. (5 points) Suppose the Wronskian of two functions f and g is W(f,g) = sin(a:). Which of the follwing statements is FALSE? (a) f and g are linearly independent on any open interval. (b) f and g are linearly independent on the interval (—g, g). i/(c) f and 9 can be solutions to a second order linear homogeneous dif- ferential equation on the interval (—g. g). (d) f and 9 can be solutions to a second order linear homogeneous dif- ferential equation on the interval (0, 7r). 2. (5 points) Which of the following is a suitable form for a particular solution y(t) to the differential equation y” + 6y' + 93; = 21342"3t + 48"“ cos(t) + 56'3‘ sin(t) ? A,B, C, D below are constants. r1+ 6r 4. 7 :0 (a) Ate's‘ + Bra-3‘ cos(t) + Ce‘3‘sin(t) (“4' 5 \$.30 0 —3t —3t -3t, - r‘ " r1 -' ’ 3 (b) At-e + Be cos(t) + Ce sm(t) 3 1 6-24: (c) (At2 + Bt)e‘3‘ + Cite-3‘ cos(t) +Dte‘3‘ sin(t) ‘ , 31'” / (d) (A123 + Bt2)e‘3‘ + 06"“ cos(t) + De‘3‘ sin(t) l0 3. ( 5 points ) A spring is stretched L meters by mass of 4 kilograms. The system is set in motion at time t = (l by an external force F(t) = sin(wt) Newtons. Assume no damping and take 9 = 10m/secz. Then resonance will occur when 4 0 4—10 :‘lzL. ”5—- \/ (a) w2L=10. Co 2 )K <4 (b)w2L=‘20. 0 7: —° ‘xF—E (c) w2L=30. “5 3‘30 ‘=‘~ .‘2 (d) sz = 40. 4. (5 points) If Y1=t. Y2=t+2e‘, Y3=t+e‘+e‘t are solutions to a nonhomogeneous differential equation 3/” + p(t)y’ + W)?! = 9(t), where W) 75 0, then the general solution to this differential equation is (a) c1t+ 02(t + 26‘) + c3(t + e‘ + e"), where cl, 02, 63 are any constants. (b) c1t+ Cge‘ + (236", where c1, c2, (:3 are any constants. (c) alt + c2(t + 2e‘) + t + et + e“: where (:1, c; are any constants. /(d) cle‘ + age“ + t, where (:1, c2 are any constants. 5. {5 points) A spring—mass system has mass 4kg and spring constant 9m/secz. What is the critical value of the damping constant 7 (the value for which the system goes from underdamped to overdamped state)? (a)'y=4kg/sec “43‘1“. k1?- (b)7=6kg/Sec 4r7'4 Yr *? =0 'l‘M Mf‘“+‘4/‘-n£ (‘7 7—, (c)'y=9kg/sec X. " (Pu-“l => .. /(d)7=12kg/sec < Y“ “‘§=I'L 6. (5 points) Which of the following CANNOT be the graph of a solution to a differential equation of the form mzr"+'ya:’+kx=0, m,7,k>0? b x (a) x ( ) t t x x t (o) t (d )l/ H:r Partial Credit Section 7. (15 points) Given that y1 = m2 is a solution to 2:231” — 2331/ + 2y = 0, a: > 0, (a) Find the second solution yg linearly independent of y1. Explain why yl and y2 are linearly independent. - grit—'L‘jl-i-L :0 x70 um: aura, 37 xii , / 1. - P(X)alx .i w(‘3"1‘3¥)¢w(93,v3’i)5v3| 2 Ceg :C‘efxﬂb’C / _ if ”V“ — C8 hx=C><1. ly‘l' 91M 3" 0m; («kccwly “Ad-{’1 we; w(3‘171.)*0 (b) Find the general solution to this differential equation. Gum/”(L \3:c.x1+ clx’ c,’ (I M 5 8.(20 points) Consider the nonhomogeneous differential equation y” — 4y"+ 5y = tea + 3. (a) Find two linearly independent solutions of the corresponding homoge- neous differential equation. \$7M Char 2 ' rl'ciNs—ﬁo, r r = L Mo‘q—iz‘ V” \, 1, .. L .. 7. L =LiL x=1, ﬁ‘l 2t 3,:eltm’c. ‘31=€,A\-t. ‘3‘, y 1.x... Axel-LP mm, M WI— “49C propmh‘cmdl arm/(5., 5940 (b) Find a particular solution of the nonhomogeneous equation. 0‘: T‘g Y(+)= (At+B-)e Ni. %V 1 x CD Y’: Ae1t+ll4tH3let =(1At+ mime“? 0 Y”: 2A€Lt+1(lAt+A+IB)eZt_e(4At+¢A+¢(3) .. ‘t ‘ 6) ‘31 (Mt WNW) ‘ (+61% 2M + AHBB + y (Ae+s)e‘t+i~c : telt+3 he, I“M449 +SC= teltlrg‘ 3 @ YR): te Jr}. 6 if Adl-WP Mlonzo 6, 4m 5/172! (c) Find the general solution of the nonhomogeneous equation. - 1t 7x ‘ 2e Q’C.£ mt+ c1e wt + ta + 3 E) CUCIMI (d) Give the form of a particular solution to the equation 4. y” — 43/ + 5y = 3e2‘ cos(t). Do NOT solve for it. Yes)" At game + (Steumgt, mo wt. 9. (20 points) A mass of 2kg stretches a spring 100m. The mass is pulled down 200m from the equilibrium position and then released with downward initial veolicity 2m/sec. Ignore air resistance and take g = 10m/sec2. g (a) Write down the diﬂerential equation governing the motion of the mass. 714:1, L:o\(. 2-(0‘—4L~0.i .L‘cZDO. LUk): F°s€l17- 0“: rte W Rm QWCi‘Lviuv-a. after txmo 7’ 1. Differential Equation: 7. u." +2 mm -= O 2. Initial Conditions: “(0) = o. 2 , u’to) = 7. (b) Determine the position of the mass at any time t. 0‘2E : iv -— 7—(0 i. ha 1 1 - I‘m: . MngAAiOt' W1=M¢0t H: C\(L.A(o~{: + C14A1~l°t u'= ~IO€,A/Cmiot + Ioclcwtot. \l -2 C 0 7. u(o)-,C -0}, (at/(MT-lOCL ' . |- one): 0‘; le0t 4- o‘LAQIOt 7%? b (c) Find the amplitude, frequency, period, and the phase of the motion. (Macon, Cmtot + 0,7. xvcwlot = R M (”f ‘ 3‘) , “Li—Lu. m R '5 k0\z>l4_(0\1)‘- ‘2 (Kl-fl = 'amPit-l'kde. it _ 1‘ 47¢». Zl X’QV‘C‘ij-l‘z: szw W i 1. Amplitude: 0. Z J: { 2. Frequency: [ o I 3. Period: % .77 4. Phase: 1 q. ((1) Find the ﬁrst time the mass crosses the equilibrium position. gut“; aim-=0. rot—5:13 tot=§+% =23 MS fi‘gzrg/Lec, ‘10. (15 points) Find the Laplace transform of the function . ' 5 if05t<6, f(t)—{t—1ift26 from deﬁnition. 5 0° e—Atdt 2p _ : 4: (:06)- iéiw} L “C” A -ﬂ‘t‘ - 6 ~a4t 1‘»... -1 e pit gr]; “SOS—e. 0H? +A’I“ \$6 U; > -f “—454 ON: e At ~ “At 6 dc“ alt we : A. Q 47/ ’ ’U’ = Q ”A 0 €223, ‘A -J‘t’ A A _ ’t ’F LE“ 0:") .6 4" g e ’6 At A—vso ’ﬂ 6 +,4 - , -éA — ~6 _ se -i+l9t-(/*">QM_§.€24+?”k 2»? V7: 3: A“ ‘A “A 241 e ‘ J, : \_. g -65, g 0 A~A>O T “L: 4. yew \ er” -64 R + Lb» R e ,4 A—w ‘Aa. — V J. 0 FL '3- : + Q‘éﬂ A>° ' Pi- A :— ' ”7"“ Hrs)? 8 )7M; ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern