I have read and understood the instructions regarding academic dishonesty:

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1 Name Final Exam MATH 6600 SPRING 08 MARK TEST 0 ON YOUR SCANTRON! Student ID Section Number (see list below 03 UNIV 03 0:30am TR Alper, Onur 04 REC 3:30pm MWF Luo, Tao 05 UNIV 03 :30pm TR Hora, Raphael 05 UNIV 7 9:00am TR Kaufmann, Birgit 053 REC 4 :00pm TR Petrosyan, Arshak 06 UNIV 0 0:30am MWF Chen, Min 06 UNIV 03 :00pm TR Alper, Onur 063 UNIV 7 :30pm MWF Phillips, Daniel 08 REC 4 :30pm TR Petrosyan, Arshak 09 REC :30pm MWF Luo, Tao UNIV 03 8:30am MWF Jin, Long 3 UNIV 03 3:00pm TR Hora, Raphael UNIV 03 9:30am MWF Jin, Long UNIV 303 :30pm MWF Ma, Zheng 4 UNIV 303 :30pm MWF Ma, Zheng 5 UNIV 003 0:30am MWF Bell, Steven 5 UNIV 9 9:30am MWF Torres, Monica 53 UNIV 7 3:00pm TR Xu, Jie 54 UNIV 7 4:30pm TR Xu, Jie 55 UNIV 7 :30am MWF Park, Moon Gyu INSTRUCTIONS:. This exam contains 0 problems, worth 5 points each. There is one correct answer for each problem.. There is a table of Laplace transforms provided at the end of the exam. 3. Please fill in your name, ID, section number and test number 0 on the scantron. 4. Work only in the space provided, or on the backside of the pages. You must show your work. 5. Mark your answers clearly on the scantron. Also circle your choice for each problem in this booklet. 6. No books, notes, calculators, phones or other electronic devices, please. ACADEMIC DISHONESTY Purdue University faculty and students commit themselves towards maintaining a culture of academic integrity and honesty. The students taking this exam are not allowed to seek or obtain any kind of help from anyone to answer questions on this test. If you have questions, consult only an instructor or a proctor. You are not allowed to look at the exam of another student. You may not compare answers with anyone else or consult another student until after you finish your exam and hand it in to a proctor or to an instructor. You may not consult notes, books, calculators, cameras, or any kind of communications devices until after you finish your exam and hand it in to a proctor or to an instructor. If you violate these instructions you will have committed an act of academic dishonesty. Penalties for academic dishonesty can be very severe and may include an F in the course. All cases of academic dishonesty will be reported to the Office of the Dean of Students. Your instructor and proctors will do everything they can to stop and prevent academic dishonesty during this exam. If you see someone breaking these rules during the exam, please report it to the proctor or to your instructor immediately. Reports after the fact are not very helpful. Anyone who is seen handling any communication device gets automatically a score of 0 for the exam and likely an F in the course I have read and understood the instructions regarding academic dishonesty: Student name: Signature:

2 . Find the solution to the initial value problem y = 4x3 + y 6, A. y = 3 x 4 + x B. y = + x 3 + x C. y = + x 4 + x D. y = 4 4x 3 + x E. y = 4x 3 + x y( =.. Let y(t be exact solution of the following initial value problem { ty y = t, t > 0, y( = 0. Let y app (t be the approximate solution of this initial value problem by using Euler s method with step size h =. Then y app (3 = A. 6 B. 4 C. 3 D. 3 E. 4

3 3. What is the largest open interval in which the solution of the initial value problem t y ln t + y = t 5 e t sin(t 4 y(3 = π is guaranteed to exist by the Existence and Uniqueness Theorem? A. (0, 4 B. (, 5 C. (4 π, 5 D. (, 4 E. (4 π, 4 4. Identify the differential equation that corresponds to the direction field on the right. A. B. C. D. E. dy dt = y + dy dt = y(y + dy dt = y + y dy dt = y y + dy dt = y3 (y + 4 y t

4 5. Solve the initial value problem for the homogeneous equation by using a substitution v = y/x. A. y = B. y = x 3 ln x x ln x C. y = x(4 ln x 3 /4 ( 8 ln x 7 D. y = x 9 8 ln x E. y = x(3 ln x /4 /4 dy dx = x4 + y 4 xy 3, y(e = e, x > 0 6. Find the solution of ty 4y = 4t 4 e t, y( = 0. A. t e t et B. t e t C. e t e D. te t et E. t e t t 4

5 7. Find the solution to the initial value problem A. 6xy + y + x = 37 B. x 3 y + x y + y + x = 6xy + y + (3x + xy + y dy dx = 0 y( = 3 C. 3x y + x y + x 3 + x + y = 4 D. x y + xy + y + x = 3 E. 3x y + xy + y = 7 8. Consider the initial value problem, y = y(y (y, y(0 = y 0. For which initial value y 0 is lim t y(t =? A B C..00 D..00 E. None of the above 5

6 9. Determine a suitable form for Y (t if the method of undetermined coefficients is to be used on y + y = t + t sin(t. A. Y (t = At + B + t(ct + D cos(t + t(et + F sin(t B. Y (t = At + B + (Ct + D cos(t + (Et + F sin(t C. Y (t = At + B + t(ct + D(cos(t + sin(t D. Y (t = t(at + B + t(ct + D cos(t + t(et + F sin(t E. Y (t = At + t(bt + C sin(t 0. Determine the values of α such that all solutions of the following equation tend to zero as t. y (α y + (α α + y = 0. A. α > B. α < 0 C. 0.5 < α < D. 0 < α < 0.5 E. α < 0.5 6

7 . Suppose u(t = 3 sin 3t+4 cos 3t is a solution to a spring-mass system. The period and the amplitude of u(t are: A. π, 5 B. 3 π, 4 C. 3 π, 5 D. 3 π, 3 E. π, 5. The function y = t is a solution of t y + 3ty + y = 0, t > 0. If we seek a second solution y = t v(t by reduction of order, then v(t could be A. t B. ln t ln C. t D. t E. t ln t 7

8 3. Which of the following is a fundamental set of solution of 4y (4 5y 9y = 0. A. {e t, e t, cos(3t/, sin(3t/} B. {e t/, e t/, cos(3t, sin(3t} C. {e 3t/, e 3t/, cos t, sin t} D. {e 3t, e 3t, cos(t/, sin(t/} E. none of the above 4. Note that y (t = t and y (t = t are solutions of the linear homogeneous differential equation t y + 3ty y = 0. Use variation of parameters to find the general solution of the nonhomogeneous differential equation t y + 3ty y = 4t + 4t. A. C t + C t t + t B. C t + C t t3 + t C. C t + C t t4 + 5 t3 D. C t + C t t + 5 t E. C t + C t t + 5 t 8

9 5. Find the inverse Laplace transform of the function A. u 3 (t (cos(t 6 + sin(t 6 B. e (4t (cos(t 3 + sin(t 3 C. u 3 (t (e 4t e 5t D. u 3 (te (4t (cos(t sin(t 6 E. u 3 (te 4t (cos(t sin(t 3 F (s = e 3s s + s 8s The solution of the initial value problem is A. y = g(t sin t + sin t cos t B. y = t g(τ sin (t τdτ + sin t cos t 0 C. y = g(t sin t + sin t cos t D. y = t g(τ sin (t τdτ + sin t cos t 0 E. y = G(s sin t + sin t cos t y + 4y = g(t; y(0 =, y (0 = 4 9

10 7. Find the Laplace transform of the function f(t = A. B. s e s s e s s C. e s s D. s e (s s E. e s s { t, 0 t <, t. 8. In the phase portrait of the system the origin is a (an x = ( 3 4 x, A. asymptotically stable node B. unstable node C. asymptotically stable spiral point D. saddle point E. asymptotically unstable spiral point 0

11 9. The general solution to can be given by A. c e 6t ( B. c e 6t ( C. c e 6t ( D. c e 6t ( E. c e 6t ( [ ( + c e 6t t [ ( + c e 6t t + c e 6t [ t + c e 6t [ t + c e 6t [ t ( 4 4 x = 8 ( ] + ( ] 0 + ( ( ] + 0 ( ( ] 4 + ( ( ] + 0 x

12 ( 3 0. The matrix A = has complex eigenvalues r = ± i. An eigenvector corresponding ( to r = + i is. The system i has one solution given by x(t = ( ( A. c e i ( +it + c e t ( ( cos t B. c e sin t cos t t sin t + c x = Ax + e sin t cos t t + ( ( ( cos t 0 C. c e cos t t + c e sin t t + e t ( ( ( D. c e i ( +it + c e + i ( it + ( ( ( cos t 0 E. c e cos t t + c e sin t t + c 3 e t ( 3 e t 4 ( e t. What is the general solution to the system? e t ( e t

13 f(t = L {F (s} F (s = L{f(t}.. e at s a 3. t n n! s n+ 4. t p (p > 5. sin at 6. cos at 7. sinh at 8. cosh at 9. e at sin bt 0. e at cos bt s Γ(p + s p+ a s + a s s + a a s a s s a b (s a + b s a (s a + b. t n e at n! (s a n+. u c (t e cs 3. u c (tf(t c e cs F (s 4. e ct f(t F (s c 5. f(ct 6. t 0 f(t τ g(τ dτ s ( s c F, c > 0 c F (s G(s 7. δ(t c e cs 8. f (n (t s n F (s s n f(0 sf (n (0 f (n (0 9. ( t n f(t F (n (s 3

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