Page 1Math 217Exam 2Name:ID:Section:This exam has 14 questions:• 12 multiple choice worth 6 points each.• 1 hand graded worth 28 points.Important:• No graphing calculators!• For the multiple choice questions, mark your answer on the answer card.• Show all your work for the written problems. You will be graded on the ease of reading yoursolution.• You are allowed a 3 × 5 note card for the exam.1. Multiply the matrices1 2 −2−3 2 05 −2 10 1 −2−1 2 14 4 0Add up all the entries in your answer and select the closest answer below.(a) −28(b) −21(c) −14(d) −7(e) 0(f) 7(g) 14(h) 21(i) 28(j) 34(k) 43Page 2Math 217Exam 22. Consider the systemX0=3 −2 0−1 3 −20 −1 3XThis system has solutions belowX1(t) = e2t221X2(t) = e4t20−1X3(t) = e6t2−21Let W (t) = W (X1, X2, X3) be the Wronskian. Find W (0). Choose the closest answer.(a) −20(b) −16(c) −12(d) −8(e) −4(f) 0(g) 4(h) 8(i) 12(j) 16(k) 20Page 3Math 217Exam 23. Consider the initial value problemx0= x2− 2y − t x(0) = 1y0= 5x − 4y + t y(0) = −2Using Runge-Kutta with step size h = 0.5 to approximate y(0.5). Choose the closest answer.(a) −2.0(b) −1.5(c) −1.0(d) −0.5(e) 0(f) 0.5(g) 1.0(h) 1.5(i) 2.0(j) 2.5(k) 3.0Page 4Math 217Exam 24. Convert the initial value problem to a first order system and use Euler’s method with step sizeh = 0.5 to approximate x(1.0)x00− 2(x0)2+ x2t = −1 x(0) = 1, x0(0) = −1Choose the closest answer.(a) 4.40(b) 4.45(c) 4.50(d) 4.55(e) 4.60(f) 4.65(g) 4.70(h) 4.75(i) 4.80(j) 4.85(k) 4.90Page 5Math 217Exam 25. Which of the following triplets of functions are linearly independent?I. f(x) = ex, g(x) = e−x, h(x) = 1II. f(x) = ex, g(x) = cos x, h(x) = sin xIII. f(x) = x, g(x) = x2, h(x) = x2− xIV. f(x) = cos2x, g(x) = sin2x, h(x) = 1(a) II only(b) IV only(c) I and II only(d) I and IV only(e) II and III only(f) I, II and III only(g) I, II and IV only(h) II, III and IV only(i) I, II, III and IVPage 6Math 217Exam 26. Consider a mass-spring with mass m = 1, c = 0 (no damping) and k = 25. An input forcesatisfies F (t) = F0sin ωt. For what value of ω, if any, is there resonance? Choose the closestanswer.(a) ω = 0(b) ω = 1(c) ω = 4(d) ω = 5(e) ω = 8(f) ω = 9(g) ω = 16(h) ω = 25(i) No resonance possible(j) None of the above7. Choose the correct form of a particular solution to the diffe rential equation below using themethod of undetermined coefficients.y(3)− y00− 4y0= sin x + x2− 5(a) yp= A sin x + B + Cx2(b) yp= A cos x + B + Cx2(c) yp= A sin x + B + Cx + Dx2(d) yp= A cos x + B + Cx + Dx2(e) yp= A cos x + B sin x + C + Dx + Ex2(f) yp= A cos x + B sin x + x(C + Dx + Ex2)(g) yp= x(A cos x + B sin x) + C + Dx + Ex2(h) yp= x(A cos x + B sin x) + x(C + Dx + Ex2)(i) yp= x(A cos x + B sin x) + C + Dx + Ex2+ F x3(j) yp= x(A cos x + B sin x) + x(C + Dx + Ex2+ F x3)(k) The method of undetermined coefficients can not be used with this differential equation orsome other answer.Page 7Math 217Exam 28. Select the differential equation that has y = 4e2x− e−2xcos 2x as a solution.(a) y(4)− 6y000+ 15x00− 17y0+ 10 = 0(b) y000+ 2y00− 16y = 0(c) y000+ 2y00− 3y0− 10y = 0(d) y000+ y00− 4y0− 4y = 0(e) y000− y00− 4y0+ 4y = 0(f) y000− 3y0+ 2y = 0(g) y000+ 2y00− y0− 2y = 0(h) y000+ 4y00+ 5y0+ 2 = 0(i) All of the above have the given solution(j) None of the above have the given solutionPage 8Math 217Exam 29. To solve non-homogeneous equations we discussed the method of undetermined coefficients andvariation of parameters. Which of the following equations does the method of undeterminedcoefficients apply?I. 3y000− 2y00+ 1y0= e2x+ cos(x2)II. xy000− x3y00+ x2y0= cos x + sin xIII. 2y000+ 5y00− πy0= x cos x + sin xIV. 4y000+ 7y0− 13y = xex2(a) I only(b) II only(c) III only(d) IV only(e) I and II only(f) I and III only(g) I and IV only(h) II and III only(i) II and IV only(j) III and IV only(k) Some other combination or none of the abovePage 9Math 217Exam 210. Solve the initial value problemy00− 4y0+ 4y = e2x; y(0) = 0, y0(0) = 0.What is y(2)?(a) 0(b) 1(c) 2(d)12e2(e) 2e2(f)14e4(g)12e4(h) e4(i) 2e4(j) 4e4Page 10Math 217Exam 211. A mass-spring system with forced oscillation satisfies the following initial value problem:x00+ 4x = sin 2t; x(0) = 0, x0(0) = 0Find x(π).(a) −2π(b) −π/2(c) −π/2 − 1/2(d) −π/4(e) 0(f)√2/2(g) π/4(h) π/2 + 1/2(i) π/2 − 1/2(j) 2πPage 11Math 217Exam 212. Consider the eigenvalue problemy00− 2y0+ λy = 0; y(0) = 0, y(1) = 0Which of the following is an eigenvalue?I. 1II. 2πIII. π2+ 1IV. 4π2+ 1(a) I only(b) II only(c) III only(d) IV only(e) I and II only(f) I and III only(g) I and IV only(h) II and III only(i) II and IV only(j) III and IV only(k) Some other combinationWRITTEN PROBLEM—SHOW YOUR WORKMath 217Exam 2Name:ID:Section:Note: You will be graded on the readability of your work. Use the back of this sheet, if necessary.13. (a) Find all eigenvalues and eigenvectors of the matrixA =2 1 00 1 22 2 2(b) Find the general solution to the systemX0= AXwhere A is defined above and X
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