MATH 251 Examination II April 6, 2015 FORM A. Name: Student Number: Section:

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1 MATH 251 Examination II April 6, 2015 FORM A Name: Student Number: Section: Thi exam ha 12 quetion for a total of 100 point. In order to obtain full credit for partial credit problem, all work mut be hown. For other problem, point might be deducted, at the ole dicretion of the intructor, for an anwer not upported by a reaonable amount of work. The point value for each quetion i in parenthee to the right of the quetion number. A table of Laplace tranform i attached a the lat page of the exam. Pleae turn off and put away your cell phone. You may not ue a calculator on thi exam. Do not write in thi box. 1 through 7: (35) 8: (12) 9: (12) 10: (15) 11: (15) 12: (11) Total:

2 1. (5 point) Evaluate the following definite integral 0 e ( 3)t co(2t) dt. (a) (b) (c) (d) 3 ( 3) ( 3)( 2 + 4) ( + 3)( 2 + 4) + 3 ( + 3) (5 point) Find the Laplace tranform L{u 3 (t)(9 t 2 )}. (a) F () = e (b) F () = e (c) F () = e 3 (d) F () = e Page 2 of 11

3 3. (5 point) Find the invere Laplace tranform of F () = (a) f(t) = e t co(2t) 3e t in(2t) (b) f(t) = e t co(2t) e t in(2t) (c) f(t) = e t co(2t) 3e t in(2t) (d) f(t) = e t co(2t) 2e t in(2t) (5 point) Let x(t) be the olution of the initial value problem [ x 3 2 α = x, x(0) = [ 0 Suppoe lim x(t) = t 0 (a) α = 2 (b) α = 3 (c) α = 12 (d) α = 18 ], find the value of α. ]. Page 3 of 11

4 5. (5 point) Conider a certain 2 2 linear ytem x = Ax, where A i a matrix of real number. Suppoe ome of it olution do not reach a limit either a t +, or a t. Then the critical point (0, 0) mut be (a) a addle point. (b) a center. (c) either a addle point or a center. (d) neither a addle point nor a center. 6. (5 point) How many critical point doe the following ytem have? x = 2 + xy y = x + 2y (a) 2 (b) 3 (c) 4 (d) More than 4. Page 4 of 11

5 7. (5 point) Given that (2, 1) i a critical point of the ytem x = 2 + xy y = x + 2y. Which of the following tatement i TRUE regarding (2, 1)? (a) It i an aymptotically table piral point. (b) It i an untable piral point. (c) It i an aymptotically table node. (d) It i an untable addle point. Page 5 of 11

6 8. (12 point) For each part below, determine whether the tatement i true or fale. To receive credit you mut (briefly) jutify each anwer. (a) L{e (2+3t) (co(t) in(t))} = 1 e 2 L{e 3t co(t)} 1 e 2 L{e 3t in(t)}. (b) Suppoe f(t) = u 3 (t) co t + u 2 (t)t + u 4 (t), then f(π) = π. (c) Suppoe f(3) = e, then L{δ(t 3) ln(f(t))} = e 3. (d) The function 2, if t < π f(t) = t 2, if π t < 3π t, if 3π t can be rewritten a f(t) = 2 + u π (t)(t 2 2) + u 3π (t)(t t 2 ). Page 6 of 11

7 9. (12 point) Conider the ytem of linear differential equation lited below. A. x 5 2 = x 0 5 B. x 0 8 = x 2 0 C. x 3 4 = x 3 2 D. x π 0 = x 0 π For each of part (a) through (f) below, write down the letter correponding to the ytem on thi lit that ha the indicated property. There i only one correct anwer to each part. However, a ytem may be re-ued for more than one part. (a) Which ytem i (neutrally) table? (b) Which ytem ha a proper node at (0, 0)? (c) Which ytem ha a addle point at (0, 0)? (d) Which ytem ha all of it olution converge to (0, 0) a t? (e) Thi ytem coefficient matrix ha only one linearly independent eigenvector. (f) Every nonzero vector i an eigenvector of thi ytem coefficient matrix. Page 7 of 11

8 10. (15 point) (a) (5 point) Suppoe L{f(t)} = What i L{e 2t t f(t)}? (b) (5 point) Rewrite the following third order linear equation into an equivalent ytem of firt order linear equation. y + e 5t y + 3y in(2t)y = t 3. (c) (5 point) Find all value of β for which the critical point (0, 0) of the linear ytem below i a (neutrally) table center. x β 9 = x 4 β Page 8 of 11

9 11. (15 point) Ue the Laplace tranform to olve the following initial value problem. y + 4y + 3y = δ(t 1) + 2u 6 (t)(t 6), y(0) = 1, y (0) = 0. Page 9 of 11

10 12. (11 point) Conider the initial value problem. x = [ (a) (9 point) Solve the initial value problem. ] [ x, x(e 2 2 ) = 5 ]. (b) (2 point) Claify the type and tability of the critical point at (0, 0). Page 10 of 11

11 f(t) = L 1 {F ()} F () = L{f(t)} e at 1 a 3. t n, n = poitive integer 4. t p, p > 1 5. in at 6. co at 7. inh at 8. coh at 9. e at in bt 10. e at co bt 11. t n e at, n = poitive integer 12. u c (t) 1 n! n+1 Γ(p + 1) p+1 a 2 + a a 2 a 2 a 2 2 a 2 b ( a) 2 + b 2 a ( a) 2 + b 2 n! ( a) n+1 e c 13. u c (t)f(t c) e c F () 14. e ct f(t) F ( c) 15. f(ct) 16. (f g)(t) = t 0 f(t τ)g(τ) dτ 1 ( ) c F c F ()G() 17. δ(t c) e c 18. f (n) (t) n F () n 1 f(0) f (n 1) (0) 19. ( t) n f(t) F (n) ()