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MA26600 Final Exam GREEN VERSION 01 NAME: PUID (10 digits): INSTRUCTOR: SECTION/TIME: • You must use a #2 pencil on the mark-sense answer sheet. • Fill in the ten digit PUID (starting with two zeroes) and your Name and blacken in the appropriate spaces. • Fill in the correct Test/Quiz number (GREEN is 01 , ORANGE is 02 ) • Fill in the four digit section number of your class and blacken the numbers below them. Here they are: 0010 TR 10:30A G Poghotanyan 0011 TR 12:00P G Poghotanyan 0022 TR 3:00P Zhaopeng Hao 0023 TR 1:30P Zhaopeng Hao 0034 TR 1:30P Ying Liang 0035 MWF 9:30A Heejin Lee 0046 MWF 1:30P Ping Xu 0047 TR 3:00P Ying Liang 0061 MWF 3:30P Christian Noack 0062 TR 12:00P Guang Yang 0071 TR 9:00A Guang Yang 0072 MWF 10:30A E Birgit Kaufmann 0083 MWF 11:30A E Birgit Kaufmann 0084 MWF 8:30A Krishnendu Khan 0095 MWF 7:30A Krishnendu Khan 0096 MWF 10:30A Ping Xu 0107 MWF 1:30P Yilong Zhang 0108 MWF 8:30A Heejin Lee 0109 MWF 2:30P Christian Noack 0110 MWF 2:30P Michelle Michelle 0111 MWF 3:30P Michelle Michelle 0112 MWF 9:30A Ping Xu 0113 MWF 11:30A Yilong Zhang • Sign the mark-sense sheet. • Fill in your name and your instructor’s name and the time of your class meeting on the exam booklet above. • There are 20 multiple-choice questions, each worth 10 points. Blacken in your choice of the correct answer in the spaces provided for questions 1–20 in the answer sheet. Do all your work on the question sheets, in addition, also Circle your answer choice for each problem on the question sheets in case your mark-sense sheet is lost. • 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, electronic devices, or papers are allowed. Use the back of the test pages for scratch paper. • Pull off the table of Laplace transforms on the last page of the exam for reference. Do not turn it in with your exam booklet at the end. 1

1. What is the largest open interval in which a solution y ( x ) to the initial value problem ( x + 2)( x − 1) y ′ + ( x − 3) y = 2 x − 5 , y (0) = 3 is guaranteed to exist? A. ( − 2 , 3) B. (5 , ∞ ) C. ( − 2 , 1) D. ( − 2 , 5) E. ( −∞ , − 2) 2. Solve the differential equation dy dx = 4 x 3 + 6 x 2 y 2 , y (1) = 3 . A. y = 3 √ 18 x 4 + 3 x 3 + 6 B. y = 3 √ 3 x 4 + 6 x 3 + 18 C. y = 3 √ x 4 + 8 x 3 + 18 D. y = 3 √ 3 x 4 + 18 x 3 + 6 E. y = 3 √ 3 x 4 + 9 x 3 + 15 2

5. Which statements below are TRUE for this autonomous equation dy dx = 3 y ( y − 4) 2 ( y + 4) 3 ? (I) y = − 4 is an asymptotically stable equilibrium solution. (II) y = 0 is an asymptotically stable equilibrium solution. (III) There are precisely three equilibrium solutions. A. All three are TRUE B. Only (II) and (III) C. Only (I) and (III) D. Only (I) and (II) E. Only (II) 6. Solve the initial value problem: y ′′ + 4 y ′ + 3 y = 0 , y (0) = 2 , y ′ (0) = − 4 . A. e − x + 1 2 e − 3 x B. e − x + e − 3 x C. e x − e − 3 x D. e x − 1 2 e − 3 x E. e − x − e − 3 x 4

7. Find all values of k for which the general solution to y ′′ + 2(1 − k ) y ′ + k 2 y = 0 has the form y = C 1 e ax cos bx + C 2 e ax sin bx , where b ̸ = 0. (Note: a is allowed to be 0.) A. k > 1 B. k = b C. 0 < k < 1 D. k < 1 4 E. k > 1 2 8. Find the trial solution for a particular solution y p of the following differential equation using the method of undetermined coefficients. y (4) − 2 y ′′′ + 10 y ′′ = t 2 + e t cos 2 t. A. At 3 + Bt 2 + Ct + De t cos 2 t + Ee t sin 2 t B. At 4 + Bt 3 + Ct 2 + Dt + E + Fe t cos 2 t + Ge t sin 2 t C. At 4 + Bt 3 + Ct 2 + Dt 2 e t cos 2 t + Et 2 e t sin 2 t D. At 4 + Bt 3 + Ct 2 + De t cos 2 t + Ee t sin 2 t E. At 2 + Bt + C + De t cos 2 t + Ee t sin 2 t 5

11. Given that y c ( x ) = c 1 x + c 2 x − 1 is a complementary solution of the nonhomogeneous second order differential equation x 2 y ′′ + xy ′ − y = 600 x 5 , use the method of variation of parameters to find its particular solution y p ( x ). A. y p ( x ) = 3 x 7 ln x B. y p ( x ) = 125 x 5 C. y p ( x ) = 25 x 6 ln x D. y p ( x ) = 25 x 5 E. y p ( x ) = 45 2 x 7 12. Transform the following system into an equivalent system of first-order differential equa- tions by letting x 1 = x, x 2 = x ′ and y 1 = y, y 2 = y ′ : x ′′ + 3 x ′ + 4 x − 2 y = 0 , y ′′ + 2 y ′ − 3 x + y = e 2 πt . A. x ′ 1 = x 2 , x ′ 2 = − 4 x 1 − 2 y 1 − 3 x 2 , y ′ 1 = y 2 , y ′ 2 = 3 x 1 − y 1 − 2 y 2 + e 2 πt B. x ′ 1 = x 2 , x ′ 2 = − 4 x 1 + 2 y 1 + 3 x 2 , y ′ 1 = y 2 , y ′ 2 = 3 x 1 − y 1 + 2 y 2 + e 2 πt C. x ′ 1 = x 2 , x ′ 2 = − 4 x 1 + 2 y 1 − 3 x 2 , y ′ 1 = y 2 , y ′ 2 = 3 x 1 − y 1 − 2 y 2 + e 2 πt D. x ′ 1 = x 2 , x ′ 2 = − 4 x 1 + 2 y 1 − 3 x 2 , y ′ 1 = y 2 , y ′ 2 = 3 x 1 + y 1 − 2 y 2 + e 2 πt E. x ′ 1 = x 2 , x ′ 2 = − 4 x 1 − 2 y 1 − 3 x 2 , y ′ 1 = y 2 , y ′ 2 = 3 x 1 + y 1 − 2 y 2 + e 2 πt 7

13. Consider the following systems x ′ = 1 − 2 2 1 x (I) x ′ = − 1 2 − 2 − 1 x (II) Which of the following statements is true about their phase plots? A. Both (I) and (II) are saddle points B. (I) spiral source and (II) spiral sink C. (I) spiral sink and (II) saddle point D. (I) spiral source and (II) saddle point E. (I) spiral sink and (II) spiral source 8

15. Find the general solution of x ′ = 1 − 4 4 9 x . A. x ( t ) = c 1 − 4 4 e 5 t + c 2 − 1 1 e 5 t B. x ( t ) = c 1 4 0 e 5 t + c 2 0 4 t e − 5 t C. x ( t ) = c 1 − 4 4 e 5 t + c 2 1 − 4 t 4 t e 5 t D. x ( t ) = c 1 (1 + t ) − 4 4 e 5 t E. x ( t ) = c 1 − 4 4 e 5 t + c 2 − 4 t + 4 e − 5 t 10

16. Find the Laplace transform of the solution to the initial value problem x ′′ − x = 5 sin 2 t, x (0) = 1 , x ′ (0) = 1 . A. 1 s − 1 + 1 ( s − 1) 2 + 5 s 2 + 4 B. 1 ( s − 1) 2 + 5 2( s 2 + 4) C. 1 s + 1 − 1 s − 1 + 2 s 2 + 4 D. 2 s + 1 + 1 s − 1 − 1 s 2 + 2 E. − 1 s + 1 + 2 s − 1 − 2 s 2 + 4 11

18. The Laplace transform of f ( t ) = Z t 0 ( t − τ ) e t − τ cos(2 τ ) dτ is A. F ( s ) = 2 ( s 2 + 1)( s + 4) 2 B. F ( s ) = s ( s − 4) 2 ( s − 1) C. F ( s ) = 1 ( s − 1)( s 2 + 4) D. F ( s ) = s ( s − 1) 2 ( s 2 + 4) E. F ( s ) = s ( s 2 + 1)( s 2 + 4) 19. Find the Laplace transform of f ( t ) = ( 0 , t < 1 , t 2 e 2 t , t ≥ 1 . A. e − s − 2 2 ( s − 2) 3 − 2 ( s − 2) 2 + 1 s − 2 B. e − s +2 2 ( s − 2) 3 + 2 ( s − 2) 2 + 1 s − 2 C. e − s +2 2 ( s − 1) 3 + 4 ( s − 1) 2 + 4 s − 1 D. 2 e − s ( s − 2) 3 E. e − 2 s +1 2 ( s − 1) 3 + 2 ( s − 1) 2 + 1 s − 1 13

20. Find the inverse Laplace transform L − 1 { F ( s ) } of the function F ( s ) = 13 s ( s 2 + 4 s + 13) . A. 2 + cos 3 t + 1 2 sin 3 t B. 3 + e − 2 t cos 3 t − e − 2 t sin 3 t C. 1 − 2 3 e − 2 t sin 3 t D. e − 2 t cos 3 t + e − 2 t sin 3 t E. 1 − e − 2 t cos 3 t − 2 3 e − 2 t sin 3 t 14