MATH 251Examination IIJuly 28, 2008Name:Student Number:Section:This exam has 9 questions for a total of 100 points. In order to obtain full credit for partialcredit problems, all work must be shown. Credit will not be given for an answer notsupported by work. The point value for each question is in parentheses to the right of thequestion number. A table of Laplace transforms is attached as the last page of the e xam.Please turn off and put away your cell phone.You may not use a calculator on this exam.1:2:3:4:5:6:7:8:9:Total:Do not write in this box.MATH 251 EXAMINATION II July 28, 20081. (5 points) Which system of first order linear equations below is equivalent to the second orderlinear equationy00+ 5y0− 6y = 0?(a)x01= x2x02= −5x1+ 6x2(b)x01= x2x02= 6x1− 5x2(c)x01= −x1x02= 6x1+ 5x2(d)x01= 5x1− 6x2x02= x22. (5 points) Suppose y(t) is the solution of the first order linear initial value problemy0+ 2y = t3e−4t, y(0) = −3.What is Y (s), the Laplace transform of y(t)?(a) Y (s) =6(s + 2)(s + 4)4−3s + 2(b) Y (s) =6(s + 2)(s + 4)4+3s + 2(c) Y (s) =3(s + 2)(s − 4)4+3s + 2(d) Y (s) =3(s + 2)(s − 4)4−3s + 2Page 2 of 9MATH 251 EXAMINATION II July 28, 20083. (13 points) Consider a mass-spring system described by the equation2u00+ 8u0+ ku = F (t), k > 0.Answer the following questions. Be sure to justify your answer. Full credit will not be givenwithout s upporting work.(a) (4 points) For what value(s), or range of values, of k would the system be criticallydamped?(b) (3 points) Suppose the spring was stretched 5 meters by the mass to its equilibriumposition. Find the value of k. You may use g = 10 as the gravitational constant.(c) (3 points) Suppose F (t) = −5 cos 2t. Give the value(s) of k, if any, such that the systemwould undergo resonance.(d) (3 points) Suppose F (t) = 0 and k = 16. Find the quasi-period of the system.Page 3 of 9MATH 251 EXAMINATION II July 28, 20084. (12 points) Find the inverse Laplace transforms of(a) (6 points)3s2− 2s + 8s3+ 4s(b) (6 points) e−3s4s + 6s2− 6s + 25Page 4 of 9MATH 251 EXAMINATION II July 28, 20085. (12 points) Rewrite the following piecewise continuous function f (t) in terms of the unit-stepfunctions. Then find its Laplace transform L{f(t)}.f(t) =1 + 2t2, 0 ≤ t < 5e−4t− t, t ≥ 5Page 5 of 9MATH 251 EXAMINATION II July 28, 20086. (14 points) Use the Laplace transform to s olve the initial value problemy00+ 6y0+ 9y = δ(t) − 2u5(t), y(0) = 0, y0(0) = −1.No credit will be given if the Laplace transform is not used to solve this problem.Page 6 of 9MATH 251 EXAMINATION II July 28, 20087. (14 points)(a) (12 points) Solve the initial value problemx0=5 −32 −2x, x(0) =81.(b) (2 points) Classify the type and stability of the critical point of this system at (0, 0).Page 7 of 9MATH 251 EXAMINATION II July 28, 20088. (12 points) In each part below, consider a certain system of two first order linear differentialequations in two unknowns, x0= Ax.(a) (4 points) Suppose one of the eigenvalues of the coefficient matrix A is r = 4i, whichhas a corresponding eigenvector1 − 3i2. Write down the system’s real-valued generalsolution.(b) (2 points) Classify the type and stability of the critical point at (0, 0) for the systemdescribed in (a).(c) (4 points) Suppose the coefficient matrix A only has one distinct eigenvalue, r = −7,which has corresponding eigenvectors both40and−13. Write down the system’sgeneral solution.(d) (2 points) Classify the type and stability of the critical point at (0, 0) for the systemdescribed in (c).Page 8 of 9MATH 251 EXAMINATION II July 28, 20089. (13 points)(a) (5 points) Find the critical points (there are 3) of the following nonlinear system.x0= (x + 1)(y − 2)y0= y(x + y)(b) (8 points) Linearize the following nonlinear system about its critical point (4, 2) and clas-sify its type and stability.x0= x2− 4y2y0= xy − 2xPage 9 of
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