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Unformatted text preview: Final Examination - Spring, 2008. Diﬀerential Equations/Linear Algebra MTH 2201 Time: 2 hours 05/02/2008 Max.Credit: 60 Answer all the questions. Make your answers precise and write legibly. Calculators are NOT allowed. 1. (a) Find the Laplace transform of the function f (t) = et cos t. (b) Find the inverse Laplace transform of F (s) = s(s2 −12s+2) (c) Use the Laplace transform to solve the initial value problem y − y = et cos t, y (0) = 0, y (0) = 0.    100 2. (a) Find the eigenvalues and eigenvectors of the matrix A = 2 2 −1 . 010 (b) Use Gauss elimination method and solve the system of equations: −2k1 − k2 + k3 = −1 −k2 + k3 = 1 (c) Find the fundamental matrix for the system of diﬀerential equations: x y z =x = 2x + 2y − z =y ++ 3. Use a b b b the b a b b properties of determinants and ﬁnd the determinant of the matrix: bb b b .  a b ba  4. Find the general solution of the Cauchy-Euler equation: 2y 3y t3 d 3 − 3t2 d 2 + 6t dy − 6y = 3 + lnt3 . dt dt dt
dy dx x+3y 3 x+ y 5. Use the method variation of parameters and solve : y + 3y + 2y = sin et .  6. Solve the initial value problem: =  7. Find conditions on the b’s that ensure that the system is consistent: x1 −2x2 −x3 = b1 −2x1 +3x2 +2x3 = b2 −4x1 +7x2 +4x3 = b3 1  ...
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