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Unformatted text preview: 642:527 FALL 2007 SUMMARY OF THE METHOD OF FROBENIUS Consider the linear, homogeneous, second order equation: y + p ( x ) y + q ( x ) y = 0 . (1) Suppose that x = 0 a regular singular point : xp ( x ) = summationdisplay n =0 p n x n , | x | < R 1 , x 2 q ( x ) = summationdisplay n =0 q n x n , | x | < R 2 , R 1 , R 2 > . Define ( r ) = r ( r- 1) + p r + q ; the indicial equation is ( r ) = 0 , roots r 1 , r 2 . Case (i). r 1 and r 2 are distinct and do not differ by an integer. There are two linearly independent solutions: y 1 ( x ) = x r 1 summationdisplay n =0 a n x n , y 2 ( x ) = x r 2 summationdisplay n =0 b n x n , a = b = 1 . (2) Case (ii). r 1 = r 2 . There is one solution y 1 ( x ) of the form given in (2), and a second solution with the form y 2 ( x ) = y 1 ( x )(ln x ) + x r 1 summationdisplay n =1 b n x n . (3) Case (iii). r 1 = r 2 + m, m a positive integer. There is one solution y 1 ( x ) as in (2), and a second solution with the form y...
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- Fall '07