Page 371 - Schaum's Outline of Theory and Problems of Advanced Calculus

362 FOURIER SERIES [CHAP. 13

2x X 4cos m m t
2
1
Ans: Uðx; tÞ¼ 1 þ þ e sin mx
m
m¼1
13.51. Give a physical interpretation to Problem 13.50.
13.52. Solve Problem 13.49 with the boundary conditions for Yðx; 0Þ and Y t ðx; 0Þ interchanged, i.e., Yðx; Þ¼ 0;
Y t ðx; 0Þ¼ 0:05xð2 xÞ, and give a physical interpretation.
3:2 X 1 ð2n 1Þ x 3ð2n 1Þ t
1
Ans: Yðx; tÞ¼ sin sin
3 4 4 2 2
n¼1 ð2n 1Þ
13.53. Verify that the boundary-value problem of Problem 13.24 actually has the solution (14), Page 357.
MISCELLANEOUS PROBLEMS
13.54. If < x < and 6¼ 0; 1; 2; ... ; prove that
sin x sin x 2 sin 2x 3 sin 3x
2 sin ¼ 1 2 2 2 þ 3 2
2
2
2
13.55. If < x < ,prove that
sinh x sin x 2 sin 2x 3 sin 3x
2 sinh þ 1 þ 2 þ 3
ðaÞ ¼ 2 2 2 3 þ 2 2
cosh x 1 cos x cos 2x
2 sinh 2 þ 1 þ 2
ðbÞ ¼ 2 2 þ 2 2
! ! !
x 2 x 2 x 2
13.56. Prove that sinh x ¼ x 1 þ 2 1 þ 2 1 þ 2

ð2 Þ ð3 Þ
! ! !
4x 2 4x 2 4x 2
13.57. Prove that cos x ¼ 1 2 1 2 1 2

ð3 Þ ð5 Þ
[Hint: cos x ¼ðsin 2xÞ=ð2 sin xÞ:Š
ffiffiffi
p
2 1 3 5 7 9 22 13 15 .. .
13.58. Show that (a)
2 ¼ 2 2 6 6 10 10 14 14 ...

ffiffiffi 4 4 8 8 12 12 16 16 ...
p
(b) 2 ¼ 4
3 5 7 9 11 13 15 17 ...
13.59. Let r be any three dimensional vector. Show that
2
2
2
2
2
2
(a) ðr iÞ þðr jÞ @ ðrÞ ; ðbÞðr iÞ þðr jÞ þðr kÞ ¼ r 2
and discusse these with reference to Parseval’s identity.
( ) 2
b 1
ð
X
13.60. If f n ðxÞg; n ¼ 1; 2; 3; ... is orthonormal in (a; b), prove that f ðxÞ c n n ðxÞ dx is a minimum when
a n¼1
ð b
f ðxÞ n ðxÞ dx
c n ¼
a
Discuss the relevance of this result to Fourier series.