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Final2006spring

Final2006spring - Final Examination Spring 2006 Dierential...

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Final Examination - Spring, 2006. Differential Equations/Linear Algebra MTH 2201 05/01/2006 Time: 2 hours Max.Credit: 50 Answer all the questions. Make your asnwers precise and write legibly. Calculators are NOT allowed. 1. Consider the following system of differential equations. y 1 = 0 y 2 = y 1 y 3 = y 1 + y 2 . (a) Write the above system in the form Y = AY where A is a 3 × 3 matrix and Y is a 3 × 1 vector. [2] (b) Find the eigenvalues and eigen vectors of the matrix A . [1+2] (c) Find a solution of the system Ap = 0 0 1 . [2] (d) Find a solution of the system Aq = 0 1 0 . [2] (e) Find three linearly independent solutions of the system Y = AY . [3] 2. Find a general solution of the system of differential equations: [8] y 1 = - 2 y 3 y 2 = y 1 + 2 y 2 + y 3 y 3 = y 1 + 3 y 3 . 3. Without directly evaluating the determinants, show: a 1 + b 1 a 1 - b 1 c 1 a 2 + b 2 a 2 - b 2 c 2 a 3 + b 3 a 3 - b 3 c 3 = - 2 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 Clearly indicate the properties of the determinants that you are used. [4] 1
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4. Find the inverse of the matrix
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