11. (a) Let F(s) = L{f(t)}, where f(t) is piecewise continuous and of exponential order on [0, ∞o). Show 2 { [ * f (7) dr} = —- P(8). Hint: Let 9₁(t) = fő ƒ (t₁)dt₁ and note that g₁ (t) = f(t). Then use Theorem 5.2.2. (b) Show that for n ≥ 2, |\x{ ſ'* ƒ˜"* ··.·. [*^*ƒ(tr) dt.1…..dtn} = ——_F(s).
11. (a) Let F(s) = L{f(t)}, where f(t) is piecewise continuous and of exponential order on [0, ∞o). Show 2 { [ * f (7) dr} = —- P(8). Hint: Let 9₁(t) = fő ƒ (t₁)dt₁ and note that g₁ (t) = f(t). Then use Theorem 5.2.2. (b) Show that for n ≥ 2, |\x{ ſ'* ƒ˜"* ··.·. [*^*ƒ(tr) dt.1…..dtn} = ——_F(s).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:11. (a) Let F(s) = L{f(t)}, where f(t) is piecewise continuous and of exponential order on [0, ∞o). Show
|£ { ["* f(7) dr} = = F(6).
Hint: Let 9₁(t) = fő ƒ (t₁)dt₁ and note that g'₁ (t) = f(t). Then use Theorem 5.2.2.
(b) Show that for n ≥ 2,
| Մ
2 { ["*
•tn
nt₂
1
"* - [*^ 1(tr)}dtir.. dtn} = = P().
... f(t1)dt₁...dtn "} -F(s).
(i)
(ii)
Transcribed Image Text:THEOREM 5.2.2
Suppose that fis continuous and f' is piecewise continuous on any interval 0 ≤ t ≤ A. Suppose further
that f and f' are of exponential order, with a as specified in Theorem 5.1.6. Then L {f'(t)} exists for
s > a, and moreover
L{f'(t)} = sL{f(t)} - f(0).
(3)
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