Chem Differential Eq HW Solutions Fall 2011 183

# Chem Differential Eq HW Solutions Fall 2011 183 - Section...

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Section A.3 Linear Ordinary Diﬀerential Equations with Nonconstant Coeﬃcients A183 17. Put the equation in standard form: xy ±± +2(1 - x ) y ± +( x - 2) y =0 ,y 1 = e x ; y ±± + 2(1 - x ) x y ± + x - 2 x y ,p ( x )= 2 x - 2; ± p ( x ) dx =2 l n x - 2 x e - ± p ( x ) dx = e - 2ln x +2 x = e 2 x x 2 ; y 2 = e x ± e 2 x x 2 e 2 x dx = e x ± 1 x 2 dx = - e x x . Hence the general solution y = c 1 e x + c 2 e x x . 21. y ±± - 4 y ± +3 y = e - x . λ 2 - 4 λ +3=0 ( λ - 1)( λ - 3) = 0 λ =1or λ =3 . Linearly independent solutions of the homogeneous equation: y 1 = e x and y 2 = e 3 x . Wronskian: W ( x ² ² ² ² ² e x e 3 x e x 3 e 3 x ² ² ² ² ² e 4 x - e 4 x e 4 x . We now apply the variation of parameters formula with g ( x e - x ; y p = y 1 ± - y 2 g ( x ) W ( x ) dx + y 2 ± y 1 g ( x ) W ( x ) dx = e x ±
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