exam3sampleans

exam3sampleans - Sample questions for Exam 3 1. Find the...

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Unformatted text preview: Sample questions for Exam 3 1. Find the general solution to y 00- 2 y- 8 y = te 4 t + 3 e t using variation of parameters. Solution: The homogeneous solution is y h = C 1 e 4 t + C 2 e- 2 t . Assume the particular solution has the form y p = u 1 ( t ) e 4 t + u 2 ( t ) e- 2 t where the unknown functions u 1 ( t ) and u 2 ( t ) satisfy u 1 e 4 t + u 2 e- 2 t = 0 u 1 (4 e 4 t ) + u 2 (- 2 e- 2 t ) = te 4 t + 3 e t . Multiplying the first equation by- 4 and adding it to the second equation eliminates u 1 :- 6 u 2 e- 2 t = te 4 t + 3 e t u 2 =- 1 6 te 6 t- 1 2 e 3 t . Multiplying the first equation by 2 and adding it to the second eliminates u 2 : 6 u 1 e 4 t = te 4 t + 3 e t u 1 = t 6 + 1 2 e- 3 t . Integrating, u 1 ( t ) = t 2 12- 1 6 e- 3 t u 2 ( t ) =- 1 36 te 6 t + 1 216 e 6 t- 1 6 e 3 t y p ( t ) = t 2 12- 1 6 e- 3 t e 4 t +- 1 36 te 6 t + 1 216 e 6 t- 1 6 e 3 t e- 2 t = t 2 12- t 36 + 1 216 e 4 t- 1 3 e t . The general solution is y ( t ) = C 1 e 4 t + C 2 e- 2 t + t 2 12- t 36 + 1 216 e 4 t- 1 3 e t . 1 2. Find the solution to y 00 + y- 2 y = 5 e 3 t , y (0) = 1 , y (0) =- 1 using (a) variation of parameters and (b) Laplace transforms. Solution: (a) Variation of parameters: y h = C 1 e t + C 2 e- 2 t y p = u 1 ( t ) e t + u 2 ( t ) e- 2 t u 1 e t + u 2 e- 2 t = 0 u 1 e t- 2 u 2 e- 2 t = 5 e 3 t Solving for u 1 and u 2 , u 1 = 5 3 e 2 t u 2 =- 5 3 e 5 t Then u 1 = 5 6 e 2 t u 2 =- 1 3 e 5 t y p = 5 6 e 3 t- 1 3 e 3 t = 1 2 e 3 t y ( t ) = C 1 e t + C 2 e- 2 t + 1 2 e 3 t . The initial conditions give C 1 + C 2 + 1 2 = 1 and C 1- 2 C 2 + 3 2 =- 1. Therefore C 1 =- 1 2 and C 2 = 1. Therefore y ( t ) =- 1 2 e t + e- 2 t + 1 2 e 3 t . (b) Laplace transforms. s 2 Y- s + 1 + sY- 1- 2 Y = 5 s- 3 Y = s s 2 + s- 2 + 5 ( s- 3)( s 2 + s- 2) = s ( s- 1)( s + 2) + 5 ( s- 3)( s- 1)( s + 2) = 1 3 1 s- 1 + 2 3 1 s + 2 + 1 2 1 s- 3- 5 6 1 s- 1 + 1 3 1 s + 2 Y =- 1 2 1 s- 1 + 1 s + 2 + 1 2 1 s- 3 . y ( t ) =- 1 2 e t + e- 2 t + 1 2 e 3 t . 2 3. Find the general solution to y 00 + 9 y = sec 3 t....
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exam3sampleans - Sample questions for Exam 3 1. Find the...

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