184_pdfsam_math 54 differential equation solutions odd

184_pdfsam_math 54 differential equation solutions odd - w...

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Chapter 4 So, a general solution is given by y ( t )= c 1 e t cos t + c 2 e t sin t, where c 1 and c 2 are arbitrary constants. To fnd the solution that satisfes the initial conditions, y (0) = 2 and y 0 (0) = 1, we frst diFerentiate the solution ±ound above, then plug in given initial conditions. This yields y 0 ( t )= c 1 e t ( cos t sin t )+ c 2 e t (cos t sin t )and y (0) = c 1 =2 , y 0 (0) = c 1 + c 2 =1 . Thus c 1 =2 , c 2 = 3, and the solution is given by y ( t )=2 e t cos t +3 e t sin t. 23. The auxiliary equation ±or this problem is r 2 4 r + 2 = 0. The roots o± this equation are r = 4 ± 16 8 2 =2 ± 2 , which are real numbers. A general solution is given by w ( t )= c 1 e (2+ 2) t + c 2 e (2 2) t ,where c 1 and c 2
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Unformatted text preview: w (0) = 0 and w (0) = 1, we frst diFerentiate the solution ound above, then plug in our initial conditions. This gives w (0) = c 1 + c 2 = 0 , w (0) = 2 + 2 c 1 + 2 2 c 2 = 1 . Solving this system o equations yields c 1 = 1 / (2 2) and c 2 = 1 / (2 2). Thus w ( t ) = 1 2 2 e (2+ 2) t 1 2 2 e (2 2) t = 2 4 e (2+ 2) t e (2 2) t is the desired solution. 180...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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