midterm2-B-solution-public

# midterm2-B-solution-public - Math 201 Fall 2011 Midterm...

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Math 201 Fall 2011 Midterm Exam 2 (B) October 28, 2011 1. [5 marks] Find the solution of the initial value problem y 00 - 2 y 0 + 2 y = 0 , y (0) = 1 , y 0 (0) = 0 The auxiliary equation λ 2 - 2 λ + 2 = 0 , λ = 1 ± i . So the two linearly independent solutions are y 1 = e x cos x , y 2 = e x sin x The general solution is y = c 1 y 1 + c 2 y 2 = c 1 e x cos x + c 2 e x sin x We use the initial values to determine c 1 and c 2 y (0) = c 1 = 1 y 0 = c 1 e x cos x - c 1 e x sin x + c 2 e x sin x + c 2 e x cos x , y 0 (0) = c 1 + c 2 = 0 , c 2 = - c 1 = - 1 So, y = e x cos x - e x sin x 2. [5 marks] Find the general solution to y 00 - y 0 - 2 y = - 6 e - x The general solution to the homogeneous part: Auxiliary equation: λ 2 - λ - 2 = 0 , λ = - 1 , λ = 2 Two linearly independent solutions: y 1 = e - x , y 2 = e 2 x Guess for a particular solution: y p = axe - x (we need x multiplied to e - x because e - x is a solution to the homogeneous part). y 0 p = a (1 - x ) e - x , y 00 p = a ( x - 2) e - x , y 00 p - y 0 p - 2 y p = a ( x - 2) e - x - a (1 - x ) e - x - 2 axe - x = - 3 ae - x = - 6 e - x - 3 a = - 6 , a = 2 So, y p = 2 xe - x The general solution is y

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