Page 343 - Schaum's Outline of Theory and Problems of Advanced Calculus

334 IMPROPER INTEGRALS [CHAP. 12

n
n
12.77. (a)If lfFðxÞg ¼ f ðsÞ,prove that lfx FðxÞg ¼ ð 1Þ f ðnÞ ðsÞ,giving suitable restrictions on FðxÞ.
2
s 1
(b) Evaluate lfx cos xg. Ans: ; s > 0
2 2
ðbÞ
ðs þ 1Þ
1
1
1
12.78. Prove that l ff ðsÞþ gðsÞg ¼ l f f ðsÞg þ l fgðsÞg, stating any restrictions.
12.79. Solve using Laplace transforms, the following differential equations subject to the given conditions.
(a) Y ðxÞþ 3Y ðxÞþ 2YðxÞ¼ 0; Yð0Þ¼ 3; Y ð0Þ¼ 0
0
00
0
(b) Y ðxÞ Y ðxÞ¼ x; Yð0Þ¼ 2; Y ð0Þ¼ 3
0
0
00
(c) Y ðxÞþ 2Y ðxÞþ 2YðxÞ¼ 4; Yð0Þ¼ 0; Y ð0Þ¼ 0
0
0
00
2
x
1
Ans. ðaÞ YðxÞ¼ 6e x 3e 2x ; ðbÞ YðxÞ¼ 4 2e x x; x
2 ðcÞ YðxÞ¼ 1 e ðsin x þ cos xÞ
12.80. Prove that lfFðxÞg exists if FðxÞ is piecewise continuous in every finite interval ½0; bŠ where b > 0 and if FðxÞ
is of exponential order as x !1, i.e., there exists a constant such that je x FðxÞj < P (a constant) for all
x > b.
12.81. If f ðsÞ¼ lfFðxÞg and gðsÞ¼ lfGðxÞg,prove that f ðsÞgðsÞ¼ lfHðxÞg where
x
ð
FðuÞGðx uÞ du
HðxÞ¼
0
is called the convolution of F and G,written F G.

ð M ð M
Hint: Write f ðsÞgðsÞ¼ lim e su FðuÞ du e sv GðvÞ dv
M!1 0 0
M M
ð ð
¼ lim e sðuþvÞ FðuÞ GðvÞ du dv and then let u þ v ¼ t:
M!1 0 0
1
12.82. (a)Find l 1 : ðbÞ Solve Y ðxÞþ YðxÞ¼ RðxÞ; Yð0Þ¼ Y ð0Þ¼ 0.
0
00
2 2
ðs þ 1Þ
x
ð
YðuÞ sinðx uÞ du. [Hint: Use Problem 12.81.]
(c) Solve the integral equation YðxÞ¼ x þ
0
x
ð
3
Ans.(a) 1 ðsin x x cos xÞ; RðuÞ sinðx uÞ du; ðcÞ YðxÞ¼ x þ x =6
2 ðbÞ YðxÞ¼
0
12.83. Let f ðxÞ; gðxÞ, and g ðxÞ be continuous in every finite interval a @ x @ b and suppose that g ðxÞ @ 0.
0
0
ð x
f ðxÞ dx is bounded for all x A a and lim gðxÞ¼ 0.
Suppose also that hðxÞ¼
a x!0
ð ð
1 1
(a)Prove that f ðxÞ gðxÞ dx ¼ g ðxÞ hðxÞ dx.
0
a a
(b)Prove that the integral on the right, and hence the integral on the left, is convergent. The result is that
ð
1
under the give conditions on f ðxÞ and gðxÞ, f ðxÞ gðxÞ dx converges and is sometimes called Abel’s
integral test. a
ð b
Hint: For (a), consider lim f ðxÞ gðxÞ dx after replacing f ðxÞ by h ðxÞ and integrating by parts. For (b),
0
b!1 a

b
ð
g ðxÞ hðxÞ dx @ HfgðaÞ gðbÞg; and then let b !1.
0
first prove that if jhðxÞj < H (a constant), then
a
ð ð
1 sin x 1
p
12.84. Use Problem 12.83 to prove that (a) dx and (b) sin x dx; p > 1, converge.
0 x 0