# 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: SubstitutingSolution CurveEigenvectorPopulationsDeterminantFundamental SetHomogeneousMultiplicityEquilibrium PointGeneralized

## Sample Document Text

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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