Page 1Math 217Final ExamName:ID:Section:This exam has 16 multiple choice questions.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.Function Transform Function Transformf0(t) sF (s) − f(0) 11sf00(t) s2F (s) − sf(0) − f0(0) t1s2Rt0f(τ) dτF (s)stnn!sn+1eatf(t) F (s − a) taΓ(a + 1)sa+1u(t − a)f(t − a) e−asF (s)1√πt1√s(f ∗ g)(t) =Zt0f(τ)g(t − τ) dτ F (s)G(s) cos ktss2+ k2tf(t) −F0(s) sin ktks2+ k21tf(t)R∞sF (σ) dσ cosh ktss2− k2f(t), period p11 − e−psZp0e−stf(t) dt sinh ktks2− k2u(t − a)e−ass12k3(sin kt − kt cos kt)1(s2+ k2)2eat1s − at2ksin kts(s2+ k2)2tneatn!(s − a)n+112k(sin kt + kt cos kt)s2(s2+ k2)2sin(A + B) = sin A cos B + cos A sin Bcos(A + B) = cos A cos B − sin A sin B2 cos A cos B = cos(A − B) + cos(A + B)2 sin A sin B = cos(A − B) − cos(A + B)2 sin A cos B = sin(A − B) + sin(A + B)Page 2Math 217Final Exam1. Solve the differential equationdydx= 3x2y − y, y(1) = 2Find y(2) and select the closest answer.(a) 0(b) 10(c) 25(d) 50(e) 100(f) 200(g) 500(h) 800(i) 1000(j) 1500(k) 25002. Solve the initial value problemy0+1xy = 2 sin(x2), y(√π) = 0What is y(2.5)?(a) −1.0(b) −0.9(c) −0.8(d) −0.7(e) −0.6(f) −0.5(g) −0.4(h) −0.3(i) −0.2(j) −0.1(k) 0.0Page 3Math 217Final Exam3. Solve the initial value problemxy2y0= x3+ y3, y(1) = 0What is y(2)?(a) −5(b) −2.5(c) 0(d) 2.5(e) 5(f) 7.5(g) 10(h) 12.5(i) 15(j) 17.5(k) 204. Solve the initial value problemyexy+ (xexy+ 1)y0= 0, y(0) = 1What is x when y = 0.5?(a) 0.6(b) 0.7(c) 0.8(d) 0.9(e) 1.0(f) 1.1(g) 1.2(h) 1.3(i) 1.4(j) 1.5(k) 1.6Page 4Math 217Final Exam5. Find the equilibrium solutions of the autonomous equationy0= y3+ y2− 2yand determine which equilibria are stable.(a) Stable: y = −2, y = 1; Unstable: y = 0(b) Stable: y = −2, y = 2; Unstable: y = 0(c) Stable: y = −1, y = 1; Unstable: y = 0(d) Stable: y = −1, y = 2; Unstable: y = 0(e) Stable: y = 0, y = 3; Unstable: y = 1(f) Stable: y = 1, y = 3; Unstable: y = 2(g) Stable: y = 0; Unstable: y = −2, y = 1(h) Stable: y = 0; Unstable: y = −2, y = 2(i) Stable: y = 0; Unstable: y = −1, y = 1(j) Stable: y = 1; Unstable: y = 0, y = 3(k) Stable: y = 2; Unstable: y = 1, y = 3Page 5Math 217Final Exam6. Use Runge-Kutta with step size h = 0.5 to estimate y(1).y0= x2+ xy2, y(0) = 1(a) 1.9(b) 2.0(c) 2.1(d) 2.2(e) 2.3(f) 2.4(g) 2.5(h) 2.6(i) 2.7(j) 2.8(k) 2.9Page 6Math 217Final Exam7. Solve the initial value problemy00− 2y0+ 2y = 0, y(0) = 0, y0(0) = 1What is y(1)?(a) 1.1(b) 1.6(c) 1.9(d) 2.3(e) 2.5(f) 2.7(g) 2.9(h) 3.1(i) 3.5Page 7Math 217Final Exam8. Which of the following are solutions to the differential equationy00+ 2y0− 3y = 2 cos x + 6 sin xI. y = −sin x −cos xII. y = ex− sin x − cos xIII. y = e−3x+ exIV. y = e−3x+ ex− sin x(a) I only(b) II only(c) IV only(d) I and II only(e) I and IV only(f) II and III only(g) I, II and III only(h) I, II and IV only(i) II, III and IV only(j) I, II, III and IV(k) Some other answerPage 8Math 217Final Exam9. For what value of α will the following mass-spring system have resonance?α2x00+ x = sin 2t(a) 0(b) 1/8(c) 1/4(d) 1/2(e) 1/√2(f)√2(g) 2(h) 4(i) 8(j) No value of α will cause resonance.10. Consider the differential equation modelling a mass-spring system.31x00+ 16x0+ 2x = 0(a) This equation is an example of resonance(b) This equation is an example of overdamping(c) This equation is an example of underdamping(d) This equation is an example of critical damping(e) None of the abovePage 9Math 217Final Exam11. Solve the systemx0= x + 2y x(0) = 2y0= 2x + y y(0) = 0What is x(1), choose the closest answer.(a) 14(b) 16(c) 18(d) 20(e) 22(f) 24(g) 26(h) 28(i) 30(j) 32(k) 34Page 10Math 217Final Exam12. Which of the following are eigenvalues of the matrixA =0 1 12 1 22 0 1(a) A has two or less eigenvalues.(b) A has exactly three eigenvalues and they are: −1, 0, 1(c) A has exactly three eigenvalues and they are: −1, 0, 2(d) A has exactly three eigenvalues and they are: −1, 0, 3(e) A has exactly three eigenvalues and they are: −1, 1, 2(f) A has exactly three eigenvalues and they are: −1, 1, 3(g) A has exactly three eigenvalues and they are: 0, 1, 2(h) A has exactly three eigenvalues and they are: 0, 1, 3(i) A has exactly three eigenvalues and they are: 0, 2, 3(j) A has exactly three eigenvalues and they are: 1, 2, 3(k) A has four or more eigenvalues.Page 11Math 217Final Exam13. Consider the initial value problemx0=tx + y, x(0) = 0y0=x + ty, y(0) = 1Use Euler’s method with h = 0.5 to estimate x(1).(a) 1.0(b) 1.1(c) 1.2(d) 1.3(e) 1.4(f) 1.5(g) 1.6(h) 1.7(i) 1.8(j) 1.9(k) 2.0Page 12Math 217Final Exam14. Let F (s) = L{f(t)} wheref(t) =e2t− 1tFind F (4), choose the closest answer.(a) 0.0(b) 0.1(c) 0.2(d) 0.3(e) 0.4(f) 0.5(g) 0.6(h) 0.7(i) 0.8(j) 0.9(k) 1.015. Let f(t) = L−1{F (s)} whereF (s) =1s2− 2s + 17Find f(2), choose the closest answer.(a) 0.0(b) 1.6(c) 1.8(d) 2.0(e) 2.2(f) 2.4(g) 2.6(h) 2.8(i) 3.0(j) 3.2Page 13Math 217Final Exam16. Which of the following equations is obtained by taking the Laplace transform of the initial valueproblemy00+ 2y0− ty = f(t), y(0) = 0, y0(0) = 0where f(t) =(1 if 1 ≤ t < 40 otherwise(a) (s2+ s)Y (s) =e−s− e−4ss(b) (s2+ s)Y (s) =e−4s− e−ss(c) (s2+ s)Y (s) = e−s− e−4s(d) (s2+ s)Y (s) = e−4s− e−s(e) Y0(s) − (s2+ 2s)Y (s) =e−s− e−4ss(f) Y0(s) + (s2+ 2s)Y (s) =e−s− e−4ss(g) Y0(s) + (s2+ 2s)Y (s) = e−s− e−4s(h) Y0(s) + (s2+ 2s)Y (s) = e−4s− e−s(i) Y0(s) + (s2+ 2s)Y (s) =es− e4ss(j) Y0(s) − (s2+ 2s)Y (s) =e−4s− e−ss(k) Y0(s) − (s2+ 2s)Y (s) = es− e4sPage 14Math 217Final Exam17. Identify the true statements for the differential equationx2(x2− 4)y00+ (x + 2)(sin x)y0+ x2(x + 2)3y = 0I. There is exactly 1 singular pointII. There are exactly 2 singular pointsIII. There are 3 or more singular pointsIV. 0 is a regular singular pointV. 0 is an irregular singular point(a) I only(b) II only(c) III only(d) I and IV only(e) I and V only(f) II and IV only(g) II and V only(h) III and IV only(i) III and V only(j) None of the abovePage 15Math 217Final Exam18. The point x = 0 is an ordinary point to the differential equation below and therefore there is aseries solution y =P∞n=0cnxn. Find the recurrence relation for the
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