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**Unformatted text preview: **Mathematics 38 Differential Equations Exam I 12:00–1:20, September 29, 2008 No calculators, notes, or books are allowed. Please make sure all electronic devices are turned off and out of sight. Show all work and cross out work you do not want graded! Remember to sign your blue book. With your signature you are pledging that you have neither given nor received assistance on this exam. Good luck! 1. (5 points) Check that 1 /t and 1 /t 2 are solutions of ( t 2 D 2 + 4 tD + 2) x = 0 for t > and decide whether those solutions form a complete collection of solutions. Solution: For 1 /t we get ( t 2 D 2 + 4 tD + 2) t − 1 = t 2 ( − 1)( − 2) t − 3 + 4 t ( − 1) t − 2 + 2 t − 1 = (2 − 4 + 2) t − 1 = 0 for all t > . For 1 /t 2 we get ( t 2 D 2 + 4 tD + 2) t − 2 = t 2 ( − 2)( − 3) t − 4 + 4 t ( − 2) t − 3 + 2 t − 2 = (6 − 8 + 2) t − 2 = 0 for all t > . They form a complete collection because the Wronskian is det parenleftbigg t − 1 t − 2 − t − 2 − 2 t...

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