Chem Differential Eq HW Solutions Fall 2011 185

# Chem Differential Eq HW Solutions Fall 2011 185 - Section...

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Section A.3 Linear Ordinary Diﬀerential Equations with Nonconstant Coeﬃcients A185 Hence the general solution y = x - 3 [ c 1 cos(2 ln x )+ c 2 sin(2 ln x )] . 41. We have y 2 = y 1 ± e - ± p ( x ) dx y 2 1 dx. Using the product rule for diﬀerentiation y ± 2 = y ± 1 ± e - ± p ( x ) dx y 2 1 dx + y 1 e - ± p ( x ) dx y 2 1 = y ± 1 ± e - ± p ( x ) dx y 2 1 dx + e - ± p ( x ) dx y 1 . So W ( y 1 ,y 2 )= ² ² ² ² ² ² y 1 y 1 ³ e - ± p ( x ) dx y 2 1 dx y ± 1 y ± 1 ³ e - ± p ( x ) dx y 2 1 dx + e - ± p ( x ) dx y 1 ² ² ² ² ² ² = y 1 y ± 1 ± e - ± p ( x ) dx y 2 1 dx + e - ± p ( x ) dx - y ± 1 y 1 ± e - ± p ( x ) dx y 2 1 dx = e - ± p ( x ) dx > 0 . This also follows from Abel’s formula, (4), Section A.1. 45. (a) From Abel’s formula (Theorem 2, Section A.1), the Wronskian is y 1 y ± 2 - y ± 1 y 2 = Ce -∫ p ( x ) dx , where y 1 and y 2 are any two solution of (2). (b) Given y 1 , set C = 1 in (a) y 1 y ± 2 - y ± 1 y 2 = e p ( x ) dx . This is a ±rst-order diﬀerential equation in
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