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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  2. (a) Use Gauss-Jordan elimination method and solve the system of equations; a +c =0 b +d = 0  4a +c =0 4b +d = 1 (b) Find the inverse transform of F (s) =
1 (s2 +1)(s2 +4)  (c) Use Laplace transform method and ﬁnd the solution of the initial value problem:  y + 4y = sin t u(t − 2π ), y (0) = 1, y (0) = 0. −1 −2 . 3 4    3 3 .   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 .  value problem, X = AX + F, X (0) = 5 4. Find a fundamental matrix of X = 2 8 −1 −2 X.  5. Use the method of undertermined coeﬃcients and ﬁnd the general solution of the second order ODE  y (4) + y = x. 1 6. Find the general solution of Cauchy-Euler equation: x2 y + xy − y = ln x.  7. Find the general solution of (4xy + 3x2 ) dx + (2y + 2x2 ) dy = 0.  The purpose of Education is concentration of mind, not mere collection of facts. 2 ...
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