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 °° 2 y ° +2 y =0 , y (0) = 1 , y ° (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 y ° = 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) = c 1 + c 2 , c 2 = c 1 = 1 So, y = e x cos x e x sin x 2. [5 marks] Find the general solution to y °° y ° 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 aso lut iontothehomogeneouspart) .
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midterm2-B-solution-public - Math 201 Fall 2011 Midterm...

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