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Chem Differential Eq HW Solutions Fall 2011 181

Chem Differential Eq HW Solutions Fall 2011 181 - Section...

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Section A.3 Linear Ordinary Differential Equations with Nonconstant Coefficients A181 Solutions to Exercises A.3 1. We apply the reduction of order formula and take all constants of integration equal to 0. y + 2 y - 3 y = 0 , y 1 = e x ; p ( x ) = 2 , p ( x ) dx = 2 x, e - p ( x ) dx = e - 2 x ; y 2 = y 1 e - p ( x ) dx y 2 1 dx = e x e - 2 x e 2 x dx = e x e - 4 x dx = e x - 1 4 e - 4 x = - 1 4 e - 3 x . Thus the general solution is y = c 1 e x + c 2 e - 3 x . 5. y + 4 y = 0 , y 1 = cos 2 x . y + 4 y = 0 , y 1 = cos 2 x ; p ( x ) = 0 , p ( x ) dx = 0 , e - p ( x ) dx = 1; y 2 = y 1 e - p ( x ) dx y 2 1 dx = cos 2 x 1 cos 2 2 x dx = cos 2 x sec 2 2 xdx = cos 2 x 1
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