Page 870 - Advanced_Engineering_Mathematics o'neil

850 Answers to Selected Problems

29. f (x) = sin(πx)
3.555896220J 1 (3.831705970x) + 1.670058301J 1 (7.015586670x)
+ 0.9956101332J 1 (10.173468144x) + 0.7772068876J 1 (13.32369194x)
+ 0.6036626350J 1 (16.47063005x)
31. f (x) = x
7.749400696J 2 (5.135622302x) − 0.1583973994J 2 (8.417244140x)
+ 1.310726377J 2 (11.61984117x) − 0.2381008476J 2 (14.79595178x)
+ 0.9524470038J 2 (17.95981949x)
33. f (x) = xe −x
1.418532841J 2 (5.135622302x) + 0.2923912667J 2 (8.417244140x)
+ 0.7581692534J 2 (11.61984117x) + 0.1399888559J 2 (14.79595178x)
+ 0.5434687461J 2 (17.95981949x)
35. f (x) = sin(πx)
3.733991576J 2 (5.135622302x) + 2.468532251J 2 (8.417244140x)
+ 1.700629359J 2 (11.61984117x) + 1.356527124J 2 (14.79595178x)
+ 1.099075410J 2 (17.95981949x)
37. With t =ry,
y
∞ ∞ x−1 1

e
r x t x−1 −rt dt =r x e −y dy
0 0 r r

∞ 1
e
=r x y x−1 −y dy
r x
0
∞

e
= y x−1 −y dy = (x),
0
with y used instead of t as the variable of integration in the last line.
39. Let t = u/(1 + u) to obtain
∞

B(x, y) = t x−1 (1 − t) y−1 dt
0
y−1
∞
x−1
u 1 1
du
1 + u 1 + u (1 + u) 2
0
∞ x−1
u

= x+y du.
0 (1 + u)
CHAPTER SIXTEEN THE WAVE EQUATION
16.1 Derivation of the Equation
1. Compute
2 2
2
2
∂ y n π c nπx nπct
=− sin cos and
∂t 2 L 2 L L
2
2
∂ y n π 2 nπx nπct
=− sin cos
∂x 2 L 2 L L
3. Compute
2
∂ y 1
= ( f (x + ct) + f (x − ct))

∂x 2 2
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October 14, 2010 17:50 THM/NEIL Page-850 27410_25_Ans_p801-866