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Unformatted text preview: Final Examination - Summer, 2007. Diﬀerential Equations/Linear Algebra MTH 2201 Time: 2 hours 07/06/2007 Max.Credit: 60 Answer all the questions. Make your answers precise and write legibly. Calculators are NOT allowed. 1. (a) Solve the following system of linear equations: a+b+c 5a + 4b + c + d 10a + 6b + +d 6a = = = = 0 0 2 1   (b) Find the inverse Laplace transform of the following function: Y (s) = 2s + 1 s(s + 1)(s2 + 4s + 6) (c) Use Laplace transform methods to solve the following initial value problem: y + 4y + 6y = 1 + et y (0) = 0, y (0) = 0 2. (a) Find a fundamental matrix Φ(t) of the system of ODE x y = 3x − y = 9x − 3y   (b) Find the inverse Φ−1 (t) of the fundamental matrix Φ(t) obtained in part (a).  1 (c) Find a particular solution ψ (t) of x y using the formula ψ (t) = Φ(t) = 3x − y + 1 = 9x − 3y + 3
t 0  Φ−1 (s)f (s) ds where f = 1 3 , and Φ(t) is the fundamental matrix obtained in (a). (d) Find the general solution of the nonhomogeneous system in (c).  3. (a) Transform the diﬀerential equation x2 y + xy + 4y = sin(2 ln x) into a diﬀerential equation with constant coeﬃcients.  (b) Find the general solution of the diﬀerential equation with constant coeﬃcients obtained in (a), using the method of undertemined coeﬃcients. (c) Use the solution obtained in (b) to ﬁnd the general solution of the diﬀerential equation in (a). 4. Compute A−1 where 2 5 5 A = −1 −1 0 . 2 4 3 5. Solve the diﬀerential equation (y 2 + ty )dt − t2 dy = 0.     2 ...
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