MA26600 FINAL EXAM INSTRUCTIONS Fall 2015

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1 MA266 FINAL EXAM INSTRUCTIONS Fall 25 NAME INSTRUCTOR. You must use a #2 pencil on the mark sense sheet (answer sheet. 2. On the mark sense sheet, fill in the instructor s name (if you do not know, write down the class meeting time and location and the course number which is MA Fill in your NAME and blacken in the appropriate spaces. 4. Fill in the SECTION Number boxes with section number of your class (see below if you are not sure and blacken in the appropriate spaces. 4 TR 2:PM Chi Li TR 9:AM Changyou Wang 52 TR :3AM Chi Li 3 TR 2:PM Thomas Sinclair 6 MWF 2:3PM Heather Lee 32 MWF :3AM Chin-Yi Chan 62 MWF 3:3PM Heather Lee 5 TR 3:3PM Xu Zhang 7 MWF :3AM Daniel Phillips 52 MWF :3AM Chin-Yi Chan 72 TR 7:3AM Rachel Davis 53 TR :3AM Samy Tindel 8 TR 9:AM Rachel Davis 54 TR 2:AM Samy Tindel 5. Leave the TEST/QUIZ NUMBER blank. 6. Fill in the -DIGIT PURDUE ID and blacken in the appropriate spaces. 7. Sign the mark sense sheet. 8. Fill in your name and your instructor s name on the question sheets (above. 9. There are 2 questions, each worth points. Blacken in your choice of the correct answer in the spaces provided for questions 2 in the answer sheet. Do all your work on the question sheets. Turn in both the mark-sense sheets and the question sheets when you are finished.. Show your work on the question sheets. Although no partial credit will be given, any disputes about grades or grading will be settled by examining your written work on the question sheets.. NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper. 2. The Laplace transform table is provided at the end of the question sheets.

2 . Let y(t be a solution to If we know that find the constant a. t 2 y + 2ty = 2at, for t >. y( =, y ( = 2, A. 2 B. C. D. E Identify the differential equation that produces the given direction field. 3 2 y t A. y = y(y 2 B. y = y(2 y C. y = y 2 (y 2 D. y = y(y 2 2 E. y = y 2 (y 2 2 2

3 3. Which of the following functions can be a solution to A. y = x(c 2 ln(x B. y = x(c + 2 ln(x C. y = x C 2 ln x D. y = x C + 2 ln x E. y = Cx 2 ln x y = x2 + y 2. xy 4. Initially, a tank contains 4 L of water with kg of salt in solution. Water containing.kg of salt per liter (L is entering at a rate of L/min, and the mixture is allowed to flow out of the tank at a rate of 2 L/min. Let Q(t be the amount of salt at time t measured in kilograms. What is the right formulation of the differential equation for Q(t? A. dq dt =. 2Q(t 4 B. dq dt =. 4 2Q(t 4 t C. dq dt = 4 2Q(t 4 + t D. dq dt E. dq dt =. 2Q(t 4 + t =. 2Q(t 4 t 3

4 5. Which one of the following statements is true about equilibrium solutions to y = 9y y 3? A. y = 3 and y = 3 are unstable, y = is asymptotically stable B. y =, y = 3, y = 3 are all unstable C. y =, y = 3, y = 3 are all asymptotically stable D. y = 3 and y = are asymptotically stable and y = 3 is unstable E. y = 3 and y = 3 are asymptotically stable and y = is unstable 6. The solution to the problem 2xy + 2xy (x 2 + 2x 2 y + 2y dy dx A. y 2 + x 2 + x 2 y + x = 8 B. y 2 x 2 + x 2 y + y 2 = C. x 2 y + y 2 x 2 + x = 7 D. x 2 y + y 2 x 2 + y 2 + x = E. x 2 y + y 2 + x = 7 = and y( = 2 is 4

5 7. Use the Euler method to find the approximate value at t = for y = 2x + y +, y( = 3 with h =.5. A. 5 B. 8.5 C. 3 D. 6.5 E On which interval is the initial value problem { (5 ty + (t 4y + 2y = ln t y(2 = 8 guaranteed to have a unique solution? A. (, 5 B. (, C. (, 5 D. (, 4 E. ( 5, 5

6 9. Find the general solution of the differential equation y y + 27y = A. y = c e 2t cos( 5t + c 2 e 2t sin( 5t B. y = c e 5t + c 2 e 2t C. y = c e 5t cos( 2t + c 2 e 5t sin( 2t D. y = c e 5t + c 2 e 2t E. y = c e 5t cos( 5t + c 2 e 5t sin( 5t. The function y = e x is a solution of (x y 2xy + (x + y =. If we seek a second solution y 2 = e x v(x by reduction of order, then v(x = A. (x 3 3 B. (x 2 2 C. (x 2 e x D. (x 3 3 E. (x 3 3 e x 6

7 . By the Method of Undetermined Coefficients, which of the following Y is the correct form of a particular solution to the equation A. Y = At + B + Ce t + De 3t B. Y = At 2 + Bt + Cte t + De 3t C. Y = At 3 + Bt 2 + Cte t + De 3t D. Y = At 3 + Bt 2 + Cte t + Dte 3t E. None of the above. y (4 3y (3 + 2y (2 = 4t e t + 3e 3t? 2. The general solution to the homogeneous equation t 2 y + 7ty + 5y = on the interval < t < is y c (t = c t + c 2 t 5. A particular solution to the inhomogeneous equation t 2 y + 7ty + 5y = t has the form y p (t = u (tt + u 2 (tt 5. Which of the following are satisfied by u and u 2? A. u = t 5, u 2 = t B. u = t 4, u 2 = t5 4 C. u = t3 4, u 2 = t7 4 D. u = t 7, u 2 = t 3 E. u = t, u 2 = t 5 7

8 3. An undamped, free vibration u + 9u = has initial conditions u( = 4, u ( = 9. The solution of this initial value problem can be written as u = R cos(ωt δ. What are R and δ? A. R = 5, δ = π 2 B. R = 6, δ = π 2 C. R = 5, δ = tan ( 3 4 D. R = 2 5, δ = tan ( 3 4 E. R = 2 5, δ = tan ( Let F (s = (s 3 and G(s = 5. What is the inverse Laplace transform of the product 3 s of F (sg(s? A. 3 B. 6 C. 3 D. 6 E. 3 t t t t t (t 2 τe 3t τ sin(5τd τ (t 2 τe 3(t τ sin(5τ dτ (t τ 2 e 3(t τ sin(5τ dτ (t τ 2 e 3(t τ sin(5τ dτ (t τ 2 e 3(t τ sin(5(t τ dτ 8

9 5. Which of the following functions has Laplace transform equal to A. 2 e t cos(2t sin(2t B. 2 e t e t cos(2t e t sin(2t C. 2 e t e t cos(2t e t sin(2t D. 3e t cos(2t e t sin(2t E. 2 e t + 3e t cos(2t e t sin(2t 3s 2 + 4s (s + (s 2 + 2s + 5? 9

10 6. Solve the initial value problem y + 4y + 5y = δ ( t π 4, y( =, y ( = 2. A. e 2t cos t + u π 4 (te π 2 2t sin(t π 4 B. e 2t cos t u π 4 (te π 2 2t sin(t π 4 C. e 2t cos t + u π 4 (te2t sin t D. e 2t cos t u π 4 (te 2t sin t E. e 2t cos t + u π 4 (te 2t sin t

11 7. The solution x 2 (t determined by the initial value problem { x = x 2 x 2 = x with initial conditions x ( =, x 2 ( = 2 is given by A. x 2 (t = 2e t + 3e t B. x 2 (t = 2 sin t + cos t C. x 2 (t = 2 e t et D. x 2 (t = 2 cos t sin t E. None of the above 8. In the phase portrait of the system x = ( x, the origin is a A. asymptotically stable spiral point B. saddle point C. asymptotically stable node D. asymptotically unstable node E. asymptotically unstable spiral point

12 9. Find the solution of the following initial value problem ( 3 6 x = 2 ( 2 A. x(t = 4 ( 2 B. x(t = ( C. x(t = 5 7 ( D. x(t = E. x(t = c ( + ( 3 3 ( 3 e t 2 e 3t e 3t + ( 6 e t ( 6 e 4t e 3t + c 2 ( 6 e 4t e 4t x. x( = ( 2

13 2. Consider the differential system: ( 2 x = 4 A set of fundamental solutions to this system is given by x. ( A. x ( = e 3t ( and x (2 = te 3t ( + e 3t B. x ( = e 3t ( ( and x (2 = te 3t ( + e 3t ( C. x ( = e 3t ( and x (2 = te 3t ( + e 3t ( ( 2 2 D. x ( = e t and x (2 = te t ( + e t E. x ( = e 3t ( 2 and x (2 = te 3t ( 2 + e 3t ( 3

MA 266 FINAL EXAM INSTRUCTIONS May 8, 2010

MA 266 FINAL EXAM INSTRUCTIONS May 8, 200 NAME INSTRUCTOR. You must use a #2 pencil on the mark sense sheet (answer sheet). 2. On the mark-sense sheet, fill in the instructor s name (if you do not know,