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Unformatted text preview: Math 527, Test #2(SAMPLE Solutions) University of New Hampshire Fall 2007 NAME: Section: PART 1  Closed Notes. Please complete any 3 of the 4 problems provided. Cross out the problem you do not want graded in the table below. In cases of ambiguity, we will grade the last three problems . This portion of the exam is closed calculator, closed notes and closed neighbor. No collaboration is permitted. If you have any questions, please raise your hand. Question Points Possible 1 20 2 20 3 20 4 20 1 1. Solve each of the following homogeneous equations for their general solutions. (a) y 00 2 y 8 y = 0 Assuming an exponential solution, y = ke rt the characteristic polynomial be comes: r 2 2 r 8 = 0 Solving this quadratic we find distinct, real roots: r = 4 , 2 . This yields the general solution: y = C 1 e 4 t + C 2 e 2 t (b) y 00 + 2 y + 10 y = 0 Assuming an exponential solution, y = ke rt the characteristic polynomial be comes: r 2 + 2 r + 10 = 0 Solving this quadratic using the quadratic formula we find complex roots: r = 2 ± √ 4 40 2 r = 1 ± 3 i This yields the general solution: y = C 1 e t cos(3 t ) + C 2 e t sin(3 t ) (c) y 00 + 6 y + 9 y = 0 Assuming an exponential solution, y = ke rt the characteristic polynomial be comes: r 2 + 6 r + 9 = 0 Solving this quadratic using the quadratic formula we find a repeated root: r = 3 . This yields the general solution: y = C 1 e 3 t + C 2 te 3 t 2 (d) t 2 y 00 + 4 ty + 2 y = 0 Here we have an Euler equation so we make the guess y = at r . Using this guess we obtain the characteristic polynomial: t 2 r ( r 1) t r 2 + 4 trt r 1 + 2 t r = 0 r ( r 1) + 4 r + 2 = 0 r 2 + 3 r + 2 = 0 Solving for the roots we find r = 2 , 1 . We obtain the general solution: y = C 1 t 2 + C 2 t 1 3 2. Solve the following homogeneous initial value problems: (a) y 00 + y = 0, y (0) = 1, y (0) = 0 The characteristic polynomial here is: r 2 + 1 = 0 We obtain pure imaginary roots, r = ± i and we use trig functions for our general solution....
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This note was uploaded on 04/08/2008 for the course MATH 527 taught by Professor Boucher during the Fall '07 term at New Hampshire.
 Fall '07
 Boucher
 Math, Differential Equations, Equations

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