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Final2005summer

# Final2005summer - Final Examination Dierential...

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Final Examination Differential Equations/Linear Algebra MTH 2201 07/08/2005 Time: 2 hours Max.Credit: 50 Answer all the questions. Make your asnwers precise and write legibly. Calculators are NOT allowed. 1. Determine whether x = 0 is a ordinary point/singular point for the following differential equations: (i) The Legendre Equation (1 - x 2 ) y - 2 xy + y = 0 . (ii) 2 x 2 y - xy + (1 + x ) y = 0 . [4] 2. Consider the matrix A = 2 - 1 3 2 . (a) Find the eigen values and eigen vectors of A . [4] (b) Find the fundamental matrix Φ( t ) of the differential system x = A x . [3] (c) Find the inverse Φ - 1 ( t ) of the fundamental matrix Φ( t ) . [3] (d) Find a particular solution of the nonhomogeneous system x = A x + g(t) where g ( t ) = e t t . [4] 3. If 1 i is an eigenvector for a matrix A, corresponding to an eigenvalue λ = - 1 2 + i, give two linearly independent real valued solutions of the sytem of

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