function f(t) 1 tn eat sin at cos at sinh at cosh at e-at f(t) U(ta) or Ua(t) (a ≥ 0) (ta) (a> 0) U(t − a)f(t − a) or Ua(t)f(t − a) f(n)(t) (−t)" f (t) - (fg)(t) = f(r)g(t − T) dr Laplace transform F(s) 1/s (s > 0) n!/sn+1 (s > 0) 1/(s – a) (s> a) a/(s²+a²) (s > 0) s/(s² + a²) (s > 0) a/(s² - a²) (s> |a|) s/(s² – a²) F(s+a (s>|a|) -as e S S 0) as e as e S F(s F(n)(s) F(s)G(s) sn ¯¹ƒ (0) — sn−² ƒ'(0) … … … — ƒ (n−¹) (0) Solve the differential equation บ'' "-y-12y=-4-5₁(t), y(0) = −2, y'(0) = 1 using Laplace transforms. The solution is y(t) and y(t): = = fort > 4 for 0 < t <4
function f(t) 1 tn eat sin at cos at sinh at cosh at e-at f(t) U(ta) or Ua(t) (a ≥ 0) (ta) (a> 0) U(t − a)f(t − a) or Ua(t)f(t − a) f(n)(t) (−t)" f (t) - (fg)(t) = f(r)g(t − T) dr Laplace transform F(s) 1/s (s > 0) n!/sn+1 (s > 0) 1/(s – a) (s> a) a/(s²+a²) (s > 0) s/(s² + a²) (s > 0) a/(s² - a²) (s> |a|) s/(s² – a²) F(s+a (s>|a|) -as e S S 0) as e as e S F(s F(n)(s) F(s)G(s) sn ¯¹ƒ (0) — sn−² ƒ'(0) … … … — ƒ (n−¹) (0) Solve the differential equation บ'' "-y-12y=-4-5₁(t), y(0) = −2, y'(0) = 1 using Laplace transforms. The solution is y(t) and y(t): = = fort > 4 for 0 < t <4
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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Question
Transcribed Image Text:function f(t)
1
tn
eat
sin at
cos at
sinh at
cosh at
e-at f(t)
U(ta) or Ua(t) (a ≥ 0)
(ta) (a> 0)
U(t − a)f(t − a) or Ua(t)f(t − a)
f(n)(t)
(−t)" f (t)
-
(fg)(t) = f(r)g(t − T) dr
Laplace transform F(s)
1/s
(s > 0)
n!/sn+1 (s > 0)
1/(s – a)
(s> a)
a/(s²+a²) (s > 0)
s/(s² + a²)
(s > 0)
a/(s² - a²)
(s> |a|)
s/(s² – a²)
F(s+a
(s>|a|)
-as
e
S S
0)
as
e
as
e
S
F(s
F(n)(s)
F(s)G(s)
sn
¯¹ƒ (0) — sn−² ƒ'(0) … … … — ƒ (n−¹) (0)
Transcribed Image Text:Solve the differential equation
บ''
"-y-12y=-4-5₁(t), y(0) = −2, y'(0) = 1
using Laplace transforms.
The solution is y(t)
and
y(t):
=
=
fort > 4
for 0 < t <4
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