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Unformatted text preview: Final Examination Differential Equations/Linear Algebra MTH 2201 Time: 2 hours 05/05/2005 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/regular or irregular singular point for the following differential equations: (i) The Bessel Equation x2 y + xy + (x2 - 2 )y = 0 where is a constant. (ii) xy + (1 - cos x)y + x2 y = 0.  2. Consider the matrix A = -2 1 1 -2 .    (a) Find the eigen values and eigen vectors of A. (b) Find the fundamental matrix (t) of the differential system x = A x. (c) Find the inverse -1 (t) of the fundamental matrix (t). (d) Find a particular solution of the nonhomogeneous system 2e-t x = A x + g(t) where g(t) = . 3t 3. Find the general solution 1 1 1 where A = 2 1 -1 -3 2 4 4. If of the first order homogeneous system x = A x .   2 + 2i is an eigenvector for a matrix A, corresponding to an eigenvalue -1 = 2i, give two linearly independent real valued solutions of the sytem of ODE x = A x.  1 5. Find the solution of the Cauchy-Euler equation t2 y - ty + 2y = 0, satisfying the initial values y(1) = 1, y (1) = 0.  6. If y1 , y2 are two linearly independent solutions of y + ay + by = 0 prove that y3 = y1 + y2 and y4 = y1 - y2 are also linearly indepedent.  7. Find the solution of the logistic equation y = y - y 2 , r > 0, k > 0, are given constants and y(0) = 2.  8. Use Laplace transform method to find the solution of the IVP y + y = sin t, y(0) = 1, y (0) = -1.  Education is not mere collection of facts, but the concentration of mind. The end (result) of Education is Character. 2 ...
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