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**Unformatted text preview: **Final Examination - Spring, 2006. Diﬀerential Equations/Linear Algebra MTH 2201 Time: 2 hours 05/01/2006 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. y1 = 0 y2 = y1 y3 = y1 + y2 . (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. 0 (c) Find a solution of the system Ap = 0 . 1 0 (d) Find a solution of the system Aq = 1 . 0 (e) Find three linearly independent solutions of the system Y = AY . 2. Find a general solution of the system of diﬀerential equations: y1 = −2y3 y2 = y1 + 2y2 + y3 y3 = y1 + 3y3 . 3. Without directly evaluating the determinants, show: a1 + b 1 a1 − b 1 c 1 a2 + b 2 a2 − b 2 c 2 a3 + b3 a3 − b3 c3 a1 b 1 c 1 = −2 a2 b2 c2 a3 b 3 c 3 [4] [1+2] [2] [2] [3] [8] Clearly indicate the properties of the determinants that you are used. 1 4. Find the inverse of the matrix A = 1 1 0 0 0 1 1 0 solution of the system Ax = b where b = 00 00 10 11 1 1 1 1 . Use this A−1 to ﬁnd a [5+1] 5. Find the general solution of the Cauchy- Euler equation 4t2 y + y = 0. [5] 6. Find a particular solution of y − 5y + 6y = e2t , using the method of variation of parameters. [7] You may answer any ONE of the following questions. 7. Solve the IVP, using Laplace Transform methods: y + y = 4 δ (t − 2π ), Here the δ is the Dirac delta function. 8. Use Laplace transform to solve the IVP y + y = f (t), y (0) = 1, y (0) = 0 where f (t) = 1, 0≤t< sin t, t ≥ π 2
π 2 [8] y (0) = 1, y (0) = 0. [8] 2 ...

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