Download Exam III for Mathematics 3113: Laplace Transforms and Linear Systems - Prof. Darryl Cullou and more Exams Mathematics in PDF only on Docsity! Mathematics 3113-005 Exam III Form A April 29, 2011 Name (please print) Instructions: Read the question carefully and make sure that you answer the question given. Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short answers. I. (14) Making use of the tables when needed, calculate the following Laplace transforms and inverse Laplace transforms: (i) L ( t6/5 − e−3t ) (ii) L(4 cos2(5t)) (you may need the identity cos2(θ) = 12(1 + cos(2θ)) (iii) L(4 cosh2(2t)) (iv) L (x′′ + 3x′ − 5x), if x(0) = 2, x′(0) = 3 (give the answer in terms of X(s), the Laplace transform of x(t)). (v) f(t) if F (s) = 11 + s s2 + 7 II. (4) For the matrix P = [ −8 −11 −2 6 9 2 −6 −6 1 ] , one of the eigenvalues is −2, and an eigenvector associated to −2 is[ 3 −2 2 ] . Use this information to write one specific solution of the first-order homogeneous linear system X ′ = PX. III. (12) For a certain homogeneous first-order linear systemX ′ = PX of three equations in three unknown functions, three linearly independent solutions are X1 = e 2t [ 1 −1 −2 ] , X2 = e 4t [ 0 1 −2 ] , and X3 = e −3t [ 1 2 1 ] . (a) Write a 3× 3 matrix whose determinant is the Wronskian of these three solutions, and calculate the deter- minant. (b) Write a general solution for the system, and use Gauss-Jordan elimination to solve the initial value problem X ′ = PX, x1(0) = 1, x2(0) = 0, x3(0) = 5 (that is, find the specific solutions x1, x2, and x3 that satisfy the IVP). IV. (6) (a) Give a specific example of three nonzero 2× 2 matrices A, B, and C for which AB = AC but B 6= C. (b) Show that if A, B, and C are 2× 2 matrices for which AB = AC and det(A) 6= 0, then B = C. V. (3) Define an eigenvalue of a matrix A, and define an eigenvector associated to that eigenvalue. You may use the version of the definitions given in class, or the version given in the book, or any equivalent statement. VI. (5) Let P be the matrix [ 2 3 2 1 ] . (a) Find the eigenvalues of P . (b) For the larger of the two eigenvalues, find an associated eigenvector. VII. (7) (a) Write the definition of L(f(t)) as an integral. (b) As you know, for a ≥ 0 the step function ua(t) is defined to be 0 for 0 ≤ t < a and 1 for a ≤ t. Use the definition to calculate that L(ua(t)) = e−as s for s > 0. Write the calculation correctly using limits, not treating infinity as a number.
