Final Exam Preparation: Matrix and Differential Equations, Exams of Linear Algebra

Download Final Exam Preparation: Matrix and Differential Equations and more Exams Linear Algebra in PDF only on Docsity! APPM 2360: Final exam 7:30am – 10:00am, May 6, 2009. • ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your student ID number, (3) recitation section (4) your instructor’s name, and (5) a grading table. • Text books, class notes, and calculators are NOT permitted. • A one-page double-sided crib sheet is allowed. • For questions 1 — 6, motivate your answers. • For question 7, brief answers with no motivation are sufficient. Problem 1: (30 points) Consider the matrix A = [ −4 3 2 1 ] . (a) (20 points) Find the eigenvalues and eigenvectors of A. (b) (10 points) Find the general solution to the system [ x′ y′ ] = A [ x y ] . Problem 2: (30 points) (a) (12 points) Determine the general solution to y′ = t2 y2. (b) (12 points) Determine the general solution to (t2 + 1) y′ + 2 t y = 0. (1) (c) (6 points) Find the solution of (1) that satisfies y(1) = 1. Problem 3: (30 points) (a) (15 points) Find the general solution to y′′ − 4 y′ + 13 y = 0. (b) (15 points) Find the general solution to y′′ − 4 y′ + 13 y = t et. Problem 4: (30 points) Consider the matrix and the vector A =   1 0 −3 1 0 1 2 −1 1 1 −1 1   , b =   1 1 3   . (a) (20 points) Find the general solution to the equation Ax = b. (b) (10 points) Find the general solution to the equation Ax = 0. Problem 5: (30 points) Consider the nonlinear system { x′ = 2 y y′ = y + x− x3 (2) (a) (10 points) Identify all equilibrium points of the system (2). (b) (15 points) Compute the Jacobian matrix at each equilibrium. Determine the geometry type (for example: saddle, spiral, center, star, etc) and stability of each equilibrium. (c) (5 points) Which graph is the direction field of the system (2). −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y (A) −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y (B) −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y (C) −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y (D)