346_pdfsam_math 54 differential equation solutions odd

346_pdfsam_math 54 differential equation solutions odd - 8...

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Chapter 6 Thus p 1 ( x ) 0, p 2 ( x )=1 / ( x x +1), p 3 ( x )=1 / x +1,and g ( x ) 0. Functions p 1 ( x )and q ( x ) are continuous on whole real line; p 3 ( x ) is defned and continuous ±or x> 1; p 2 ( x )is defned and continuous ±or x> 1and x 6 = 0. There±ore, all these ±unction is continuous on ( 1 , 0) and (0 , ). The initial point lies on (0 , ), and so, by Theorem 1, the given initial value problem has a unique solution on (0 , ). 7. Assume that c 1 , c 2 ,and c 3 are constants ±or which c 1 e 3 x + c 2 e 5 x + c 3 e x 0o n( −∞ , ) . (6.1) I± we show that this is possible only i± c 1 = c 2 = c 3 = 0, then linear independence will ±ollow. Evaluating the linear combination in (6.1) at x =0 , x =ln2 ,and x = ln 2, we fnd that constants c 1 , c 2 ,and c 3 satis±y c 1 + c 2 + c 3 =0 , 8 c 1 +32 c 2 + 1 2 c 3 =0 , 1 8 c 1 + 1 32 c 2 +2 c 3 =0 . This system is a homogeneous system o± linear equations whose determinant ± ± ± ± ± ± ± ± 11 1 83 2 1 / 2 1 / 81
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Unformatted text preview: 8 1 / 32 2 = 32 1 / 2 1 / 32 2 8 1 / 2 1 / 8 2 + 8 3 2 1 / 8 1 / 32 = 2827 64 6 = 0 . Hence it has the unique trivial solution, that is, c 1 = c 2 = c 3 = 0. 9. Let y 1 = sin 2 x , y 2 = cos 2 x , and y 3 = 1. We want to fnd c 1 , c 2 , and c 3 , not all zero, such that c 1 y 1 + c 2 y 2 + c 3 y 3 = c 1 sin 2 x + c 2 cos 2 x + c 3 1 = 0 , or all x in the interval ( , ). Since sin 2 x + cos 2 x = 1 or all real numbers x , we can choose c 1 = 1, c 2 = 1, and c 3 = 1. Thus, these unctions are linearly dependent. 342...
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