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38-2E-08S-solutions

38-2E-08S-solutions - Mathematics 38 Exam II 1(5 points...

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Mathematics 38 Differential Equations Exam II March 10, 2008 1. (5 points) Determine whether the system x = - ty - z + t y = - x t - z t + 1 z = x - ty is linear. If it is linear a. determine whether it is homogeneous, b. determine its order, and c . write it in matrix form. Solution: Yes. a. no. b. 3. c. x y z = 0 - t - 1 - 1 /t 0 - 1 /t 1 - t 0 x y z + t 1 0 or vectorx = 0 - t - 1 - 1 /t 0 - 1 /t 1 - t 0 vectorx + t 1 0 . 2. (5 points) Given the o.d.e. ( D 2 - 1) x = t (N) a. find the equivalent system ( S N ) , b. the general solution of (N) is x ( t ) = c 1 e t + c 2 e t - t . You do not need to verify this . Use the general solution of (N) to obtain each component of the general solution of ( S N ) , c . write ( S N ) in matrix form, d. write the general solution of ( S N ) in the form vectorx = c 1 vector h 1 ( t ) + · · · + c n vector h n ( t ) + vector p ( t ) . Solution: a. braceleftbigg x 1 = x 2 x 2 = x 1 + t . b. braceleftbigg x 1 ( t ) = c 1 e t + c 2 e t - t x 2 ( t ) = - c 1 e t + c 2 e t - 1 .

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38-2E-08S-solutions - Mathematics 38 Exam II 1(5 points...

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