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Past Exam for MATH 211 - ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA at Rice (Rice)

Exam Information

Material Type:Final
Professor:Staff
Class:MATH 211 - ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA
Subject:Mathematics
University:Rice University
Term:Spring 2001
Keywords:
  • Substituting
  • Solution Curve
  • Eigenvector
  • Populations
  • Determinant
  • Fundamental Set
  • Homogeneous
  • Multiplicity
  • Equilibrium Point
  • Generalized
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Math 211 Final Exam - Answers Spring 2001 1. Consider the differential equation y primeprime + 5y prime + 4y = 0. a) (4 points) Find a fundamental set of solutions. Answer: The characteristic polynomial is p(?) = ? 2 +5?+4 = (?+1)(?+4). The characterisitic roots are ?1 and ?4, so the functions y 1 (t) = e ?t and y 2 (t) = e ?4t form a fundamental set of solutions. b) (3 points) What is the general solution? Answer: The general solution is y(t)= C 1 y 1 (t)+C 2 y 2 (t) = C 1 e ?t +C 2 e ?4t . c) (3 points) Find the solution satisfying y(0) = 1 and y prime (0) = 2. Answer: We need to find C 1 and C 2 so that 1 = y(0) = C 1 +C 2 2 = y prime (0) =?C 1 ? 4C 2 These equations are solved by C 1 = 2 and C 2 =?1. The solution is y(t)= 2e ?t ?e ?4t . 2. Consider the differential equation x prime = (1 +x 2 ) cos t. a) (4 points) What is the general solution? Answer: The equation is separable. Separating variables and then integrating we get in...

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