Chem Differential Eq HW Solutions Fall 2011 75

Chem Differential - Section 4.8 Bessel Series Expansions 75 13 Use(6 with p = 4 Then = 8 J4 x J3 x x = J5 x 86 J3(x J2(x J3(x(by(6 with p = 3 xx =

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Section 4.8 Bessel Series Expansions 75 13. Use (6) with p = 4. Then J 5 ( x )= 8 x J 4 ( x ) - J 3 ( x ) = 8 x ± 6 x J 3 ( x ) - J 2 ( x ) ² - J 3 ( x ) (by (6) with p =3) = ³ 48 x 2 - 1 ´ J 3 ( x ) - 8 x J 2 ( x ) = ³ 48 x 2 - 1 ´³ 4 x J 2 ( x ) - J 1 ( x ) ´ - 8 x J 2 ( x ) (by (6) with p =2) = ³ 192 x 3 - 12 x ´ J 2 ( x ) - ³ 48 x 2 - 1 ´ J 1 ( x ) = 12 x ³ 16 x 2 - 1 ´± 2 x J 1 ( x ) - J 0 ( x ) ² - ³ 48 x 2 - 1 ´ J 1 ( x ) (by (6) with p =1) = - 12 x ³ 16 x 2 - 1 ´ J 0 ( x )+ ³ 384 x 4 - 72 x 2 +1 ´ J 1 ( x ) . 17. (a) From (17), A j = 2 J 1 ( α j ) 2 µ 1 0 f ( x ) J 0 ( α j x ) xdx = 2 J 1 ( α j ) 2 µ c 0 J 0 ( α j x ) = 2 α 2 j J 1 ( α j ) 2 µ j 0 J 0 ( s ) sds (let α j x = s ) = 2 α 2 j J 1 ( α j ) 2 J 1 ( s ) s j 0 = 2 cJ 1 ( α j ) α j J 1 ( α j ) 2 . Thus, for 0 <x< 1, f ( x · j =1 2 cJ 1 ( α j ) α j J 1 ( α j ) 2 J 0 ( α j x ) . (b) The function f is piecewise smooth, so by Theorem 2 the series in (a) converges to f ( x ) for all 0 1, except at x = c , where the series converges to the average value f ( c +)+ f ( c - ) 2 = 1 2 . 21. (a) Take m =1 / 2 in the series expansion of Exercise 20 and you’ll get
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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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