MATH 251 Examination II November 5, 2018 FORM A. Name: Student Number: Section:

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1 MATH 251 Examination II November 5, 2018 FORM A Name: Student Number: Section: This exam has 14 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. Please turn off and put away your cell phone. You may not use a calculator on this exam. Do not write in this box. 1 through 9: (45) 10: (4) 11: (10) 12: (14) 13: (15) 14: (12) Total:

2 1. (5 points) Evaluate the following definite integral 0 e (1 s)t sin(3t) dt. (a) 3 s 2 + 2s + 10 (b) e s 3 s (c) (d) s 1 (s 1) s 2 2s (5 points) Find the Laplace transform L{u 2 (t)(5 4t + t 2 )}. (a) F (s) = e 2s 3s2 4s + 2 s 3 (b) F (s) = e 2s 17s2 8s + 2 s 3 (c) F (s) = e 2s 5s2 4s + 2 s 3 (d) F (s) = e 2s s2 + 2 s 3 Page 2 of 11

3 3. (5 points) Find the inverse Laplace transform of F (s) = (a) f(t) = 4e 3t cos(2t) + 2e 3t sin(2t) (b) f(t) = 4e 3t cos(2t) + 6e 3t sin(2t) (c) f(t) = 4e 3t cos(2t) (d) f(t) = 4e 3t cos(2t) 12e 3t sin(2t) 4s s 2 6s (5 points) Rewrite the following function using step functions: t 2, 0 t < 2 e 3t + cos(t), 2 t < 5 f(t) = 0, 5 t < 7 7 t, 7 t (a) (t 2)u 2 (t) + (e 3t + cos(t))(u 5 (t) u 2 (t)) + (7 t)u 7 (t) (b) (t 2)(1 u 2 (t)) + (e 3t + cos(t))(u 2 (t) u 5 (t)) + (7 t)u 7 (t) (c) (t 2)(1 u 2 (t)) + (e 3t + cos(t))(u 2 (t) u 5 (t)) + (7 t)(u 7 (t) u 5 (t)) (d) (t 2)u 2 (t) + (e 3t + cos(t))(u 5 (t) u 2 (t)) + (7 t)(u 5 (t) u 7 (t)) Page 3 of 11

4 5. (5 points) Suppose L{f(t)} = F (s), for some function f(t). What is the Laplace transform L{t 2 f (t)}? (a) F (s) + 2F (s) (b) F (s) 2F (s) (c) sf (s) + 2F (s) (d) F (s) 6. (5 points) Which linear differential equation below is equivalent to the following system of linear equations? { x 1 = 5x 1 + 7x 2 + 2t t 2 x 2 = x 1 (a) y 5y + 7y = 2t t 2 (b) y + 5y 7y = 2t t 2 (c) y 7y + 5y = 2t t 2 (d) y + 7y 5y = 2t t 2 Page 4 of 11

5 7. (5 points) Consider a certain 2 2 linear system, x = Ax, where A is a matrix of real numbers. Suppose one of the [ eigenvalues of the coefficient matrix A is r = 3 2i, which has a 1 corresponding eigenvector. What is the system s real-valued general solution? 4 + i (a) C 1 e 3t [ (b) C 1 e 3t [ (c) C 1 e 3t [ (d) C 1 e 3t [ sin 2t 4 cos 2t + sin 2t cos 2t 4 cos 2t + sin 2t sin 2t 4 cos 2t sin 2t cos 2t 4 cos 2t sin 2t + C 2 e 3t [ + C 2 e 3t [ + C 2 e 3t [ + C 2 e 3t [ cos 2t 4 sin 2t + cos 2t sin 2t 4 sin 2t cos 2t cos 2t 4 sin 2t + cos 2t sin 2t 4 sin 2t cos 2t 8. (5 points) Consider a certain 2 2 linear system x = Ax, where A is a matrix of real numbers. Suppose ALL of its solutions reach a limit as t. Then the critical point (0, 0) cannot be (a) a saddle point. (b) an improper node. (c) unstable. (d) a spiral point. Page 5 of 11

6 9. (5 points) Given that (2, 1) is a critical point of the system { x = x 2 4y 2 y = x 2 + y 2 5 Linearize this system about (2, 1) to determine the critical point is an (a) unstable node. (b) asymptotically stable node. (c) unstable saddle point. (d) unstable spiral point. 10. (4 points) Consider the system in the question #9 above. There are other critical points besides (2, 1). Find all of them. Page 6 of 11

7 11. (10 points) For each part below, determine whether the statement is true or false. You must justify your answers. (a) Given that cos(2t) = 2 cos 2 (t) 1, it follows that L{cos 2 (t)} = 1 ( s 2 s ). s (b) L{δ(t π)e t π cos(t)} = e πs. (c) Suppose f(t) = (1 u 2 (t)) sin(t) + t 2 u 4 (t), then f( π 2 ) = 1. (d) The system of linear equations x = [ [ x + has a critical point at [ 1. 2 (e) The system of linear equations x = unstable improper node. [ [ x has a critical point which is an Page 7 of 11

8 12. (14 points) In parts (a) through (e), determine the type and stability of the critical point at (0, 0) for each of the 2 2 linear systems x = Ax whose general solutions are given below. For the type, give the actual name. For the stability, use the letter A if the point is asymptotically stable, U if it is unstable, S if it is (neutrally) stable. (a) C 1 e 5t [ C 2 e 5t [ 2 3 Type Stability (b) C 1 e t [ 3 cos t 2 sin t + C 2 e t [ 2 sin t 3 cos t (c) C 1 e [ 3t 1 4 [ + C 2 e 3t 4 1 (d) C 1 e 2πt [ C 2 e 2πt [ 6t + 1 4t 2 (e) C 1 [ 4 cos 3t 4 sin 3t + C 2 [ sin 3t cos 3t (f) Which solution above represents a system having a coefficient matrix A such that every nonzero vector is one of its eigenvectors? (g) Which solution above represents a system that is equivalent to the second order equation y + 9y = 0? Page 8 of 11

9 13. (15 points) Use the Laplace transform to solve the following initial value problem. y + 4y + 5y = 2δ(t 3) u π(t)e t+π, y(0) = 1, y (0) = 0. Page 9 of 11

10 14. (12 points) Consider the system of linear equations [ x 2 2 = 2 1 x. (a) (8 points) Find the general solution of this system. (b) (2 points) Classify the type and stability of the critical point at (0, 0). (c) (2 points) Given that x(0) = [ 10 β [ 0, and lim x(t) = t 0, find the value of β. 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)