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Unformatted text preview: Final Examination - Summer, 2008. Diﬀerential Equations/Linear Algebra MTH 2201 Time: 2 hours 07/03/2008 Max.Credit: 100 Answer all the questions. Make your answers precise and write legibly. Calculators are NOT allowed. 1. (a) Write the function f (t) = cos 4t, 0 ≤ t < π in terms of unit step 0, t≥π functions and ﬁnd the Laplace transform of f . 
s . (s2 +16)2 (b) Find the inverse Laplace transform of F (s) =   (c) Use Laplace transform and solve the IVP y + 16y = f (t), y (0) = 0, y (0) = 1, where f (t) is as given in part (a). 2. Find a fundamental matrix consisting of two real valued solution vectors of the system  x y = x−y = 5x − 3y (a − 1) −1 −1 −2 (a − 1) 1 = 0. 8 5 (a + 3) 3. (a) Find the values of a for which  (b) Find the fundamental matrix for the system of diﬀerential equations:  x y z 4. Use a b b b the b a b b = x+y+z = 2x + y − z = −8x − 5y − 3z properties of determinants and ﬁnd the determinant of the matrix: bb b b .  a b ba  5. Find the general solution of the Cauchy-Euler equation: 2y 2t2 d 2 + 5t dy + y = (t2 − t). dt dt 6. Use the method of variation of paramaters and ﬁnd the general solution of y + 3y + 2y = sin et .  1 7. (a) Show that the Laplace transform of the solution y (t) of the initial value problem: y + 4y + 6y = 1 + e−t , y (0) = 0, y (0) = 0 is given by 2 Y (s) = s(s+1)(s+1 s+6) .  s2 +4 (b) Use Cramer’s rule and ﬁnd the solution of the system of equations: a+b+c =0 5a + 4b + c + d = 0 10a + 6b + d = 2 6a = 1  (c) Find the inverse Laplace transform of Y (s) and hence ﬁnd the solution of the IVP in part (a).  8. Find the value of b for which the DE (xy 2 + bx2 y )dx + (x + y )x2 dy = 0 is exact and solve it using that value of b.  Education is for the concentration of mind not for mere collection of facts. 2 ...
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