Q5_1130 - MATH 415.01 QUIZ 5A Feb 10, 2010 SHOW ALL YOUR...

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Unformatted text preview: MATH 415.01 QUIZ 5A Feb 10, 2010 SHOW ALL YOUR WORK! GOOD LUCK! NAME: 1. (6 pts.) Find the Wronskian of two solutions of the given differential equation below without solving the equation. t 2 y 00- t ( t + 2) y + ( t + 2) y = 0 Rewrite the equation: y 00- t +2 t y + t +2 t 2 y = 0 W = ce- R- t +2 t dt = ce R t +2 t dt = ce R (1+ 2 t ) dt = ce t +2 ln ( t ) = ct 2 e t 2. (6.5 pts.) Find the solution of the given initial value problem below. y 00 + 2 y + 2 y = 0 , y ( / 4) = 2 , y ( / 4) =- 2 Characteristic Equation: r 2 + 2 r + 2 = 0 . roots are- 1 + i and- 1- i . Then, the solution is: y ( t ) = c 1 e- t cos t + c 2 e- t sin t Also, we have y ( t ) = c 1 (- e- t cos t- e- t sin t ) + c 2 (- e- t sin t + e- t cos t ) Use initial conditions: y ( / 4) = e- / 4 ( c 1 cos / 4 + c 2 sin / 4) = 2 / 2 e- / 4 ( c 1 + c 2 ) = 2 y ( / 4) = e- / 4 ( c 1 (- cos / 4- sin / 4) + c 2 (- sin / 4 + cos / 4)) = e- / 4 c 1 (- 2) =- 2 The last equation implies that c 1 =...
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This note was uploaded on 07/26/2011 for the course MATH 415 taught by Professor Costin during the Spring '07 term at Ohio State.

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Q5_1130 - MATH 415.01 QUIZ 5A Feb 10, 2010 SHOW ALL YOUR...

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