MATH 251Examination IIJuly 25, 2011FORM AName:Student Number:Section:This exam has 10 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.You may not use a calculator on this exam. Please turn off and put away yourcell phone and all other mobile devices.1:2:3:4:5:6:7:8:9:10:Total:Do not write in this box.MATH 251 EXAMINATION II July 25, 20111. (6 points) Find the general solution of the linear equationy(5)+ 2y(4)+ 5y000= 0.2. (5 points) Rewrite the following third order linear equation into an equivalent system of firstorder linear equations.y000+ 3y00− 2y0+ 4y = sin 2tPage 2 of 9MATH 251 EXAMINATION II July 25, 20113. (8 points)(a) (4 points) Evaluate the following definite integralZ∞0e(4−s)tcos(3t) dt.(Hint: Use the fact that this integral represents the Laplace transform of a certain function.Avoid computing it directly.)(b) (4 points) Suppose L{f (t)} =2s2s4+ 100. Use properties of the Laplace transform to de-termine L{e−πtf(t)}.4. (5 points) Suppose the linear system x0=2 − α200 −2α − 1x has an unstable propernode at (0, 0). Determine all possible value(s) of α.Page 3 of 9MATH 251 EXAMINATION II July 25, 20115. (12 points) For each part below, consider a certain system of two first order linear differentialequations in two unknowns, x0= Ax, where A is a 2x2 matrix of real numbers. Based solelyon the information given in each part, determine the type and stability of the system’s criticalpoint at (0, 0).(a) Eigenvalues of A are −1 and −6.(b) Eigenvalues of A are 3 + 7i and 3 − 7i.(c) Eigenvalues of A are 9i and −9i.(d) Eigenvalues of A are√11 and π2.(e) The general solution is x(t) = C1e−5t1−1+ C2e−5t10.(f) The general solution is x(t) = C1et−11+ C2et−t−√3 + t.Page 4 of 9MATH 251 EXAMINATION II July 25, 20116. (14 points) Find the inverse Laplace transform of each function given below.(a) (7 points) F (s) =s2+ 11s − 31(s + 5)(s2+ 36)(b) (7 points) F (s) = e−s−5s + 2s2+ 4s + 20Page 5 of 9MATH 251 EXAMINATION II July 25, 20117. (12 points) Rewrite the following piecewise continuous function f (t) in terms of the unit-stepfunction.f(t) =2t2− e−6t, 0 ≤ t < 49t + 3, 4 ≤ t.Then find its Laplace transform.Page 6 of 9MATH 251 EXAMINATION II July 25, 20118. (14 poins) Consider the initial value problemy00+ 4y = δ(t − π) + u10(t), y(0) = 1, y0(0) = −4.(a) (12 points) Use the Laplace transform to solve the initial value problem.(b) (1 point) Evaluate y(π2).(c) (1 point) Evaluate y(2π).Page 7 of 9MATH 251 EXAMINATION II July 25, 20119. (12 points)(a) (10 points) Solve the initial value problemx0=−2 41 1x, x(0) =−23.(b) (2 points) Classify the type and stabiliy of the critical point at (0, 0).Page 8 of 9MATH 251 EXAMINATION II July 25, 201110. (12 points) Consider the nonlinear system:x0= (x + 1)(y − 2) = xy − 2x + y − 2y0= (x + y)(2x − y) = 2x2+ xy − y2(a) (4 points) The system has 4 critical points. One of the critical points of the system is(−1, 1). Find the other 3 critical points of the system.(b) (4 points) Linearize the system about the critical point (−1, 1). Identify the coefficientmatrix of the linearized system.(c) (4 points) What are the eigenvalues of the coefficient matrix? Classify the type and sta-bility of the critical point at (−1, 1) by examining the linearized system found in (b).Page 9 of
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