Chem Differential Eq HW Solutions Fall 2011 175

# Chem Differential Eq HW Solutions Fall 2011 175 - Section...

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Section A.2 Linear Ordinary Differential Equations with Constant Coefficients A175 25. Equation: y - 4 y + 3 y = e 2 x ; Homogeneous equation: y - 4 y + 3 y = 0; Characteristic equation: λ 2 - 4 λ + 3 = 0 ( λ - 1)( λ - 3) = 0; Characteristic roots: λ 1 = 1 , λ 2 = 3; Solution of homogeneous equation: y h = c 1 e x + c 2 e 3 x . To find a particular solution, we apply the method of undetermined coefficients. Accordingly, we try y p = Ae 2 x ; y p = 2 Ae 2 x ; y p = 4 Ae 2 x . Plug into the equation y - 4 y + 3 y = e 2 x : 4 Ae 2 x - 4(2 Ae 2 x ) + 3 Ae 2 x = e 2 x - Ae 2 x = e 2 x ; A = - 1 . Hence y p = - e 2 x and so the general solution y g = c 1 e x + c 2 e 3 x - e 2 x . 29. Equation: y - 4 y + 3 y = xe - x ; Homogeneous equation: y - 4 y + 3 y = 0; Characteristic equation:
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