Chem Differential Eq HW Solutions Fall 2011 69

Chem Differential Eq HW Solutions Fall 2011 69 - Section...

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Section 4.7 Bessel’s Equation and Bessel Functions 69 where a 0 ± = 0 and b 0 ± = 0, and k may or may not be 0. Plugging this into the differential equation x 2 y ±± + xy ± +( x 2 - 9) y =0 and using the fact that y 1 is a solution, we have y 2 = ky 1 ln x + ± m =0 b m x m - 3 y ± 2 = ± 1 ln x + k y 1 x + ± m =0 ( m - 3) b m x m - 4 ; y ±± 2 = ±± 1 ln x + k y ± 1 x + k xy ± 1 - y 1 x 2 + ± m =0 ( m - 3)( m - 4) b m x m - 5 = ±± 1 ln x +2 k y ± 1 x - k y 1 x 2 + ± m =0 ( m - 3)( m - 4) b m x m - 5 ; x 2 y ±± 2 + xy ± 2 x 2 - 9) y 2 = kx 2 y ±± 1 ln x kxy ± 1 - 1 + ± m =0 ( m - 3)( m - 4) b m x m - 3 + kxy ± 1 ln x + 1 + ± m =0 ( m - 3) b m x m - 3 +( x 2 - 9) ² 1 ln x + ± m =0 b m x m - 3 ³ = k ln x ´ =0 µ ¶· ¸ x 2 y ±± 1 + xy ± 1 x 2 - 9) y 1 ¹ +2 kxy ± 1 + ± m =0 ´ ( m - 3)( m - 4) b m m - 3) b m - 9 b m ¹ x m - 3 + x 2 ± m =0 b m x m - 3 =2 kxy ± 1 + ± m =0 ( m - 6) mb m x m - 3 + ± m =0 b m x m - 1 . To combine the last two series, we use reindexing as follows
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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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