MATH 251 Examination II April 3, 2017 FORM A. Name: Student Number: Section:

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1 MATH 251 Examination II April 3, 2017 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown. For other problems, points might be deducted, at the sole discretion of the instructor, for an answer not supported by a reasonable amount of work. The point value for each question is in parentheses to the right of the question number. A table of Laplace transforms is attached as the last page of the exam. You may not use a calculator on this exam. Please turn off and put away your cell phone and all other mobile devices. Do not write in this box. 1 through 6: (30) 7: (8) 8: (12) 9: (13) 10: (15) 11: (12) 12: (10) Total:

2 1. (5 points) Evaluate the following definite integral 0 t 3 e (s 2)t dt. (a) (b) (c) (d) 6 (s 2) 4 3 (s 2) 3 3 (s + 2) 3 6 (s + 2) 4 2. (5 points) Find the inverse Laplace transform of F (s) = e πs 2s + 1 s 2 + 2s (a) f(t) = u π (t)(2e t+π cos(3t 3π) 1 3 e t+π sin(3t 3π)) (b) f(t) = u π (t)(2e t+π cos(3t 3π) + e t+π sin(3t 3π)) (c) f(t) = u π (t)(2e t π cos(3t 3π) + e t π sin(3t 3π)) (d) f(t) = u π (t)(2e t π cos(3t 3π) et π sin(3t 3π)) Page 2 of 11

3 3. (5 points) Find the Laplace transform L{u 4 (t)(t 2) 2 }. 4s 2 12s + 36s2 (a) F (s) = e s 3 4s 2 4s + 4s2 (b) F (s) = e s 3 4s 2 4s + 4s2 (c) F (s) = e s 4 4s 2 + 4s + 4s2 (d) F (s) = e s 3 4. (5 points) Rewrite the following function using step functions: 2t 2 e 6t, t < 4 f(t) = 9t + 3, 4 t < 10. sin(2t), 10 t (a) f(t) = (2t 2 e 6t )u 4 (t) + (9t + 3)(u 10 (t) u 4 (t)) + sin(2t)(1 u 10 (t)) (b) f(t) = (2t 2 e 6t )(1 u 4 (t)) + (9t + 3)(u 10 (t) u 4 (t)) + sin(2t)u 10 (t) (c) f(t) = (2t 2 e 6t )(1 u 4 (t)) + (9t + 3)(u 4 (t) u 10 (t)) + sin(2t)u 10 (t) (d) f(t) = (2t 2 e 6t )u 4 (t) + (9t + 3)(u 4 (t) u 10 (t)) + sin(2t)(1 u 10 (t)) Page 3 of 11

4 5. (5 points) Which ordinary differential equation below is equivalent to the following system of linear equations? { x 1 = x 2 x 2 = 3x 1 x 2 + cos t (a) u 3u + u = cos t (b) u + 3u + u = cos t (c) u + u + 3u = cos t (d) u + u 3u = cos t 6. (5 points) Consider a certain system of linear equations x = Ax, where A is a 2 2 matrix of real numbers. Suppose one of [ the ] eigenvalues of the coefficient matrix A is r = 2 i, which 1 has a corresponding eigenvector. What is the system s real-valued general solution? i (a) x(t) = C 1 e 2t cos t + C sin t 2 e 2t sin t cos t (b) x(t) = C 1 e 2t cos t + C sin t 2 e 2t sin t cos t (c) x(t) = C 1 e 2t cos t + C sin t 2 e 2t sin t cos t (d) x(t) = C 1 e 2t cos t + C sin t 2 e 2t sin t cos t Page 4 of 11

5 7. (8 points) For each part below, determine whether the statement is true or false. You must justify your answers. (a) Suppose L{f(t)} = s4 s 5 + 6, then L{e 8t f(t)} = (s + 8)4 (s + 8) (b) Suppose L{f(t)} = s4 s and given f(0) = 1, then L{f (t)} = 6 s (c) Suppose f(t) = (t 2 + 4)u e (t) 7u π (t), then f(4) = 20. (d) The system of linear equations x = is a saddle point. 1 0 x has a critical point at, which 2 1 Page 5 of 11

6 8. (12 points) Consider the systems of linear differential equations listed below. A. x 0 2 = x 2 0 B. x 1 2 = x 0 1 C. x 3 0 = x 0 3 D. x 1 3 = x 4 2 Before you start, determine the eigenvalue(s) of each system above, and write them down in the space beside each system. For each of parts (a) through (f) below, write down the letter corresponding to the system on this list that has the indicated property. There is only one correct answer to each part. However, a system may be re-used for more than one part. (a) Which system has a saddle point at (0, 0)? (b) Which system has an improper node at (0, 0)? (c) Which system is (neutrally) stable? (d) Which system has all of its solutions converge to (0, 0) as t? (e) Which system has all of its nonzero solutions diverge to infinity as t? (f) Which system has all of its solutions bounded, both as t approaches and? Page 6 of 11

7 9. (13 points) (a) (7 points) Find the inverse Laplace transform of F (s) = e 2s s s(s + 1) 2. (b) (6 points) Find the Laplace transform L{(3t + π 2 )e 2t sin(t)}. Page 7 of 11

8 10. (15 points) Use the Laplace transform to solve the following initial value problem. y 3y 4y = u 2 (t) 2δ(t 6), y(0) = 0, y (0) = 2. Page 8 of 11

9 11. (12 points) (a) (8 points) Find the general solution of the system of linear equations x = [ ] x. (b) (2 points) Find the solution satisfying x(0) = [ 3 0 ]. (c) (2 points) Classify the type and stability of the critical point at (0, 0). Page 9 of 11

10 12. (10 points) Consider the autonomous nonlinear system: x = x 2 2xy y = y + xy 1. (a) (3 points) The system has 3 critical points. One of the critical points is (0, 1). Verify that (0, 1) is indeed a critical point. Then find the other 2 critical points. (b) (7 points) Linearize the system about the point (0, 1). Classify the type and stability of this critical point by examining the linearized system. Be sure to clearly state the linearized system s coefficient matrix and its eigenvalues. Page 10 of 11

11 f(t) = L 1 {F (s)} F (s) = L{f(t)} e at 1 s a 3. t n, n = positive integer 4. t p, p > 1 5. sin at 6. cos at 7. sinh at 8. cosh at 9. e at sin bt 10. e at cos bt 11. t n e at, n = positive integer 12. u c (t) 1 s n! s n+1 Γ(p + 1) s p+1 a s 2 + a 2 s s 2 + a 2 a s 2 a 2 s s 2 a 2 b (s a) 2 + b 2 s a (s a) 2 + b 2 n! (s a) n+1 e cs 13. u c (t)f(t c) e cs F (s) s 14. e ct f(t) F (s c) 15. f(ct) 16. (f g)(t) = t 0 f(t τ)g(τ) dτ 1 ( s ) c F c F (s)g(s) 17. δ(t c) e cs 18. f (n) (t) s n F (s) s n 1 f(0) f (n 1) (0) 19. ( t) n f(t) F (n) (s)