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Mathematic Methods HW Solutions 57

# Mathematic Methods HW Solutions 57 - 2 f 2 = xe-x 4(2-x 2 f...

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Chapter 12 57 21.1 y = Ax + B ( x sinh - 1 x - x 2 + 1 ) 21.2 y = A (1 + x ) + Bxe 1 /x 21.3 y = A (1 - 2 x ) + B (1 + 2 x ) e - x 21.4 y = Ax - Be x 21.5 y = A ( x - 1) + B [( x - 1) ln x - 4] 21.6 y = A x + B [ x ln x + x ] 21.7 y = A x 1 - x + B [ x 1 - x ln x + 1+ x 2 ] 21.8 y = A ( x 2 + 2 x ) + B [( x 2 + 2 x ) ln x + 1 + 5 x - x 3 / 6 + x 4 / 72 + · · · ] 21.9 y = Ax 2 + B [ x 2 ln x - x 3 + x 4 / (2 · 2!) - x 5 / (3 · 3!) + x 6 / (4 · 4!) + · · · ] 21.10 y = Ax 3 + B ( x 3 ln x + x 2 ) 22.4 H 0 ( x ) = 1 H 1 ( x ) = 2 x H 2 ( x ) = 4 x 2 - 2 H 3 ( x ) = 8 x 3 - 12 x H 4 ( x ) = 16 x 4 - 48 x 2 + 12 H 5 ( x ) = 32 x 5 - 160 x 3 + 120 x 22.13 L 0 ( x ) = 1 L 1 ( x ) = 1 - x L 2 ( x ) = (2 - 4 x + x 2 ) / 2! L 3 ( x ) = (6 - 18 x + 9 x 2 - x 3 ) / 3! L 4 ( x ) = (24 - 96 x + 72 x 2 - 16 x 3 + x 4 ) / 4! L 5 ( x ) = (120 - 600 x + 600 x 2 - 200 x 3 + 25 x 4 - x 5 ) / 5! Note : The factor 1 /n ! is omitted in most quantum mechanics books but is included as here in most reference books. 22.20 L k 0 ( x ) = 1 L k 1 ( x ) = 1 + k - x L k 2 ( x ) = 1 2 ( k + 1)( k + 2) - ( k + 2) x +
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Unformatted text preview: 2 f 2 = xe-x/ 4 (2-x 2 ) f 3 = xe-x/ 6 (3-x + x 2 18 ) 22.30 R n =-x n Dx-n , L n = x-n-1 Dx n +1 23.6 P 1 2 n +1 (0) = (2 n + 1) P 2 n (0) = (-1) n (2 n + 1)!! 2 n n ! 23.9 For n ≤ l , i 1-1 xP l ( x ) P n ( x ) dx = 2 l (2 l-1)(2 l + 1) , n = l-1 , otherwise 23.18 (a) y = Z ( e x ) (b) y = Z p ( e x 2 / 2) 23.23 T = 1, T 1 = x , T 2 = 2 x 2-1 23.30 π/ 6...
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