Chem Differential Eq HW Solutions Fall 2011 201

Chem Differential Eq HW Solutions Fall 2011 201 - Section...

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Section A.6 The Method of Frobenius A201 Plug into xy ±± +(1 - x ) y ± + y =0 : xy ±± 1 ln x +2 y ± 1 - 1 x y 1 + ± m =0 m ( m - 1) b m x m - 1 +(1 - x ) y ± 1 ln x + y 1 x (1 - x )+(1 - x ) ± m =0 mb m x m - 1 + y 1 ln x + ± m =0 b m x m - 2 - 1 x (1 - x )+ ± m =1 m ( m - 1) b m x m - 1 + (1 - x ) 2 x + ± m =0 mb m x m - 1 - ± m =0 mb m x m + ± m =0 b m x m - 3+ x + ± m =0 [( m +1) mb m +1 +( m b m +1 - mb m + b m ] x m - x + ± m =0 [( m 2 b m +1 - m ) b m ] x m For the constant term, we get b 1 + b 0 - 3=0 . Take b 0 = 0. Then b 1 = 3. For the term in x , we get 1+2 b 2 b 2 - b 1 + b 1 b 2 = - 1 4 . For all m 3, b m +1 = m - 1 ( m 2 b m . Then b 3 = 1 9 ( - 1 4 )= - 1 36 ; b 4 = 2 16 ( - 1 36 - 1 288 ; . . . y 2 = - 3 x - 1 4 x 2 - 1 36 x 3 + ··· 17. For the equation x 2 y ±± +4 xy ± +(2 - x 2 ) y , p ( x 4 x ,x p ( x )=4 ,p 0 =4 ; q ( x 2 - x 2 x 2 2 q ( x )=2 - x 2 ,q 0 =2 . p ( x ) and q ( x ) are not analytic at 0. So
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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