04spfinsam - d x 1 d t = x 1 + 2 x 2 d x 2 d t = 3 x 1-4 x...

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Math 320 Sample Final Exam May 11, 2004, 7:45 am, B239 Van Vleck M´arton Bal´azs 1. (60 points) Solve d x 1 d t = x 1 - x 2 d x 2 d t = x 1 + x 2 , x 1 (0) = 1 x 2 (0) = - 1 . 2. (50 points) Determine the natural modes and the corresponding frequencies of the following mass-spring system: / / / / / \\\\\\\\\\ ////////// 3 / / / / / / / / / / / / / / / / / / / / / / / \\\\\\\\\\ ////////// 1 12 3 (Numbers above the springs represent the spring constants, and numbers in the bodies stand for the masses.) 3. (60 points) Use fundamental matrices and variation of parameters to solve
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Unformatted text preview: d x 1 d t = x 1 + 2 x 2 d x 2 d t = 3 x 1-4 x 2 +1 , x 1 (0) = 1 x 2 (0) = 0 . 4. (40 points) Solve the initial value problem d y d x + y x = sin x, y ( / 2) = 0 . 5. (30 points) Compute the Wronskian of ln( x ) , ln( x 2 ), and ln( x 3 ) to decide if they are linearly dependent or not on the interval (0 , ). Can you give an explanation of your result?...
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