MATH 251Examination IINovember 8, 2010FORM AName:Student Number:Section:This exam has 12 questions for a total of 100 points. In order to obtain full credit forpartial credit problems, all work must be shown. For other problems, points might bededucted, at the sole discretion of the instructor, for an answer not supported by areasonable amount of work. The point value for each question is in parentheses to the right ofthe 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.1thru6:7:8:9:10:11:12:Total:Do not write in this box.MATH 251 EXAMINATION II November 8, 20101. (5 points) Consider the nonhomogeneous second order linear equationy00+ 25y = 3e5t− 2t sin 5t.Which function below is the most suitable choice of the form of particular solution Y that youwould use to solve the given equation using the Method of Undetermined Coefficients?(a) Y = Ae5t+ (Bt + C) cos 5t + (Dt + E) sin 5t(b) Y = Ate5t+ (Bt + C) cos 5t + (Dt + E) sin 5t(c) Y = Ae5t+ (Bt2+ Ct) cos 5t + (Dt2+ Et) sin 5t(d) Y = Ate5t+ (Bt2+ Ct) cos 5t + (Dt2+ Et) sin 5t2. (5 points) Consider the fourth order linear equationy(4)+ 9y00= 0.What is its general solution?(a) y(t) = C1et+ C2e−t+ C3cos 3t + C4sin 3t(b) y(t) = C1cos√3t + C2sin√3t + C3t cos√3t + C4t sin√3t(c) y(t) = C1cos t + C2sin t + C3cos 3t + C4sin 3t(d) y(t) = C1+ C2t + C3cos 3t + C4sin 3tPage 2 of 10MATH 251 EXAMINATION II November 8, 20103. (5 points) Find the Laplace transform L{u4(t) (t − 2)2}.(a) F (s) = e−4s4s2+ 4s + 2s3(b) F (s) = e−4s4s2− 4s + 2s4(c) F (s) = e−4s4s2− 4s + 2s3(d) F (s) = e−4s36s2− 12s + 2s44. (5 points) Evaluate the following definite integralZ∞0e−(s+1)tsin(2t) dt.(Hint: This integral represents the Laplace transform of a certain function. It is absolutely notnecessary to integrate in order to find the answer.)(a)2s2+ 2s + 5(b) e−s2e−1s2+ 4(c) e−sss2+ 4(d)1s2− 2s + 5Page 3 of 10MATH 251 EXAMINATION II November 8, 20105. (5 points) Which system of first order linear equations below is equivalent to the second orderlinear equationy00− 5y0+ 6y = t2− t?(a)x01= x2x02= −5x1+ 6x2+ t2− t(b)x01= x2x02= 5x1− 6x2− t2+ t(c)x01= x2x02= −6x1+ 5x2+ t2− t(d)x01= x2x02= 6x1− 5x2− t2+ t6. (5 points) Consider a certain system of two first order linear differential equations in twounknowns, x0= Ax, where A is a matrix of real numbers. Suppose one of the eigenvaluesof the coefficient matrix A is r = 1 + 4i, which has a corresponding eigenvector−3 − 7i2.What is the system’s real-valued general solution?(a) x(t) = C1et−3 cos 4t − 7 sin 4t2 cos 4t+ C2et−7 cos 4t + 3 sin 4t2 sin 4t(b) x(t) = C1et−3 cos 4t + 7 sin 4t2 cos 4t+ C2et−7 cos 4t − 3 sin 4t2 sin 4t(c) x(t) = C1et−3 cos 4t + 7 sin 4t−2 sin 4t+ C2et7 cos 4t − 3 sin 4t2 cos 4t(d) x(t) = C1et−3 cos 4t − 7 sin 4t2 sin 4t+ C2et7 cos 4t + 3 sin 4t2 cos 4tPage 4 of 10MATH 251 EXAMINATION II November 8, 20107. (8 points) Consider various mass-spring systems and the differential equations that describetheir displacement. A list of equations is given below. Each equation may or may not describethe displacement of any mass-spring system.A. y00+ 4y0+ 5y = 0B. y00+ 4y0+ 3y = 0C. y00− y = cos tD. y00+ 9y = 0E. y00− y0− 2y = 0F. y00+ 2y0+ y = 0G. y00+ 16y = 2 sin 4tFor each of parts (a) through (d) below, write down the letter corresponding to the equationon the list above describing the correct mass-spring system with the specified behavior. Thereis only one correct equation to each part. However, an equation may be re-used for more thanone part.(a) (2 points) This system is critically damped.(b) (2 points) This system is undergoing resonance.(c) (2 points) This system has solutions that oscillate freely with a constant amplitude.(d) (2 points) This system has solutions that oscillate freely with an exponentially decreasingamplitude.Page 5 of 10MATH 251 EXAMINATION II November 8, 20108. (10 points) Determine the type and stability of the critical point at (0, 0) for each of the 2x2linear systems x0= Ax whose general solutions are given below. For the type, give the actualname. For the stability, use the letter A if the point is asymptotically stable, U if it is unstable,S if it is (neutrally) stable.Type Stability(a) C1e2tcos t2 sin t+C2e2t2 sin t−cos t(b) C1e−t11+ C2e−t−11(c) C1e√5t11+ C2e√7t−11(d) C12 cos tsin t+ C2sin t−2 cos t(e) C1e√2t11+ C2e√2tt + 1t − 1Page 6 of 10MATH 251 EXAMINATION II November 8, 20109. (14 points) Find the inverse Laplace transform of each function given below.(a) (7 points) F (s) =s2+ 1s(s + 1)2(b) (7 points) F (s) = e−9s−s + 3s2+ 6s + 25Page 7 of 10MATH 251 EXAMINATION II November 8, 201010. (16 points) Use the Laplace transform to solve the following initial value problem.y00+ 4y0+ 3y = δ(t) − u6(t), y(0) = 2, y0(0) = 0.No credit will be given if the Laplace transform is not used to solve this problem.Page 8 of 10MATH 251 EXAMINATION II November 8, 201011. (12 points) Consider the system of linear equationsx0=4 −27 −5x.(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) =−4β, and limt→∞x(t) =00, find the value of β.Page 9 of 10MATH 251 EXAMINATION II November 8, 201012. (10 points) Consider the nonlinear system:x0= x − yy0= 4y − x2y(a) (2 points) One of the critical points of the system is (2, 2). Verify that (2, 2) is indeed acritical point. That is, show that (2, 2) satisfies the condition(s) of being a critical point.(b) (2 points) Besides (2, 2), there are 2 other critical points. Find those other 2 critical pointsof the system.(c) (6 points) Linearize the system about the point (2, 2). Classify the type and stability ofthe critical point at (2, 2) by examining the linearized system. Be sure to clearly state thelinearized system’s matrix and its eigenvalues.Page 10 of
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