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Unformatted text preview: and uniqueness theorem? (d) Solve dx dt =x t2 . How many solutions satisfy x (2) = 1? (e) Why this does not violate the existence and uniqueness theorem? 3. (18 points) Find all real numbers α such that (a) t α is a solution of ( t 2 D 23 tD + 2) x = 0. (b) 2 α + 2 αt + αt 2 is a solution of ( D1) x = t 2 . Exam Continues on Other Side 1 4. (20 points) Solve xx = e t sin t , x (0) = 1. 5. (20 points) (a) Compute det e t sin t cos t e t cos tsin t e tsin tcos t (b) Use a) to show that c 1 e t + c 2 sin t + c 3 cos t is the general solution of ( D1)( D 2 + 1) x = 0 6. (20 points) (a) Find the general solution of ( D 32 D 2 + D ) x = 0. (b) Find the solution for which x (0) = x (0) = 0, x 00 (0) = 1. End of Exam 2...
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 Fall '10
 Hasselblatt
 Differential Equations, Topology, Real Numbers, Equations, Trigraph, dt dt, Tufts University Department of Mathematics

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