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Mathematic Methods HW Solutions 56

Mathematic Methods HW Solutions 56 - Chapter 12 56 10.5...

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Chapter 12 56 10.4 sin θ 10.5 sin θ (35 cos 3 θ - 15 cos θ ) / 2 10.6 15 sin 2 θ cos θ 11.1 y = b 0 cos x/x 2 11.2 y = Ax - 3 + Bx 3 11.3 y = Ax - 3 + Bx 2 11.4 y = Ax - 2 + Bx 3 11.5 y = A cos(2 x 1 / 2 ) + B sin(2 x 1 / 2 ) 11.6 y = Ae - x + Bx 2 / 3 [1 - 3 x/ 5 + (3 x ) 2 / (5 · 8) - (3 x ) 3 / (5 · 8 · 11) + · · · ] 11.7 y = Ax 2 (1 + x 2 / 10 + x 4 / 280 + · · · ) + Bx - 1 (1 - x 2 / 2 - x 4 / 8 - · · · ) 11.8 y = A ( x - 1 - 1) + Bx 2 (1 - x + 3 x 2 / 5 - 4 x 3 / 15 + 2 x 4 / 21 + · · · ) 11.9 y = A (1 - 3 x 6 / 8 + 9 x 12 / 320 - · · · ) + Bx 2 (1 - 3 x 6 / 16 + 9 x 12 / 896 - · · · ) 11.10 y = A [1 + 2 x - (2 x ) 2 / 2! + (2 x ) 3 / (3 · 3!) - (2 x ) 4 / (3 · 5 · 4!) + · · · ] + Bx 3 / 2 [1 - 2 x/ 5 + (2 x ) 2 / (5 · 7 · 2!) - (2 x ) 3 / (5 · 7 · 9 · 3!) + · · · ] 11.11 y = Ax 1 / 6 [1 + 3 x 2 / 2 5 + 3 2 x 4 / (5 · 2 10 ) + · · · ] + Bx - 1 / 6 [ x + 3 x 3 / 2 6 + 3 2 x 5 / (7 · 2 11 ) + · · · ] 11.12 y = e x ( A + Bx 1 / 3 ) 15.9 5 - 3 / 2 16.1 y = x - 3 / 2 Z 1 / 2 ( x ) 16.2 y = x 1 / 2 Z 1 / 4 ( x 2 ) 16.3 y = x - 1 / 2 Z 1 (4 x 1 / 2 ) 16.4 y = x 1 / 6 Z 1 / 3 (4 x 1 / 2 ) 16.5 y = xZ 0 (2 x ) 16.6 y = x 1 / 2 Z 1 ( x 1 / 2 ) 16.7 y = x - 1 Z 1 / 2 ( x 2 / 2) 16.8 y = x 1 / 2 Z 1 / 3 ( 2 3 x 3 / 2 ) 16.9 y = x 1 / 3 Z 2 / 3 (4 x ) 16.10 y = xZ 2 / 3 (2 x 3 / 2 ) 16.11 y = x - 2 Z 2 ( x ) 16.12 y = x 1 / 4 Z 1 / 2 ( x ) 16.14 y = Z 3 (2 x ) 16.15 y = Z 2 (5 x ) 16.16 y = Z 1 (4 x ) 16.17 y = Z 0 (3 x ) 17.7 (a) y = x 1 / 2 I 1 (2 x 1 / 2 ) (b) y = x 1 / 2 I 1 / 6 ( x 3 / 3) Note that the factor i is not needed since any multiple of y is a solution. 17.9 d dx [ x p I p ( x )] = x p I p - 1 ( x ) d dx [ x - p I p ( x )] = x - p I p +1 ( x ) I p - 1 ( x ) - I p +1 ( x ) = 2 p x I p ( x ) I p - 1 ( x ) + I p +1 ( x ) = 2 I prime p ( x ) I prime p ( x ) = - p x I p ( x ) + I p - 1 ( x ) = p x I p ( x ) + I p +1 ( x ) 18.9 Amplitude increases; outward swing takes longer. 18.10 y = ( Ax + B ) 1 / 2 J 1 / 3
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