Final2009summer

Final2009summer - Final Examination - Summer, 2009....

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Unformatted text preview: Final Examination - Summer, 2009. Differential Equations/Linear Algebra MTH 2201 Time: 2 hours 07/09/2009 Max.Credit: 60 Answer all the questions. Make your answers precise and write legibly. Calculators are NOT allowed. The numbers indicated at the end of each question are the maximum credit for the corresponding question. 1. Find the inverse Laplace transform of F (s) = 2s − 1 . s2 (s+1)3 [5] 2. (a) Use Gauss-Jordan elimination method and solve the system of equations; a +c =0 b +d = 0 [3] 4a +c =0 4b +d = 1 (b) Find the inverse transform of F (s) = 1 (s2 +1)(s2 +4) [3] (c) Use Laplace transform method and find the solution of the initial value problem: [4] y + 4y = sin t u(t − 2π ), y (0) = 1, y (0) = 0. −1 −2 . 3 4 [4] [4] [3] 3 3 . [4] [2] 3. (a) Find the eigen values and eigen vectors of A = (b) Find a fundamental matrix Φ(t) of the first order system of ODE X = AX. (c) Find the inverse Φ−1 (t) of the above fundamental matrix Φ(t). (d) Consider the nonhomogeneoeus system X = AX + F where F = Find the particular solution ψp (x) = Φ(x) x 0 Φ−1 (s)F ds. (e) Write the general solution of X = AX + F. (f) Use the general solution found in (e) and find the solution of the initial −4 . [3] value problem, X = AX + F, X (0) = 5 4. Find a fundamental matrix of X = 2 8 −1 −2 X. [7] 5. Use the method of undertermined coefficients and find the general solution of the second order ODE [5] y (4) + y = x. 1 6. Find the general solution of Cauchy-Euler equation: x2 y + xy − y = ln x. [8] 7. Find the general solution of (4xy + 3x2 ) dx + (2y + 2x2 ) dy = 0. [5] The purpose of Education is concentration of mind, not mere collection of facts. 2 ...
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This note was uploaded on 03/02/2011 for the course MATH 1101 taught by Professor Tenali during the Spring '11 term at FIT.

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Final2009summer - Final Examination - Summer, 2009....

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