Solve the initial value problem below using the method of Laplace transforms. y" - 2y' - 8y = 0, y (0) = 4, y´ (0) = 22 y(t) = (please an answer in terms of e.)
Solve the initial value problem below using the method of Laplace transforms. y" - 2y' - 8y = 0, y (0) = 4, y´ (0) = 22 y(t) = (please an answer in terms of e.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Solve the initial value problem below
using the method of Laplace transforms.
y" - 2y² - 8y = 0, y (0)= 4, y´(0) = 22
y(t) =
(please an answer in terms of e.)
Table of Laplace Transforms
F(s) = L{f(t)}
1
f(t)
1
t
eat
t", n=1,2,...
sin kt
cos kt
S
S
1
1
2
S
S>0
1
s-a
n!
n+1
S>0
"
k
2
s²+k²
S
2
s+k
2
S>0
S>0
"
2'
S>0
S>0
Properties of
Laplace Transforms
L{f+g} = L{f}+ L{g}
L{cf} = c£{f} for any constant c
L{f'} (s) = s£{f}(s)-f(0)
Lf''(s) = s²L{f}(s)-sf(0) - f'(0)
L {f(n)} (s) = s^£{f}(s) -sn-¹f(0)-sn-²₁'(0)--(n-1)(0)
£¯¹ {F₁+F₂} = £¯¹ {F₁} + £¯¹ {F2}
L-¹{cF} = CL-¹ {F}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46890c96-3b8c-4ea6-bb6b-a8358fcd3c49%2Fd83eec01-9ab7-4056-b818-df8fdfbb89a7%2Fyct8zx4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Solve the initial value problem below
using the method of Laplace transforms.
y" - 2y² - 8y = 0, y (0)= 4, y´(0) = 22
y(t) =
(please an answer in terms of e.)
Table of Laplace Transforms
F(s) = L{f(t)}
1
f(t)
1
t
eat
t", n=1,2,...
sin kt
cos kt
S
S
1
1
2
S
S>0
1
s-a
n!
n+1
S>0
"
k
2
s²+k²
S
2
s+k
2
S>0
S>0
"
2'
S>0
S>0
Properties of
Laplace Transforms
L{f+g} = L{f}+ L{g}
L{cf} = c£{f} for any constant c
L{f'} (s) = s£{f}(s)-f(0)
Lf''(s) = s²L{f}(s)-sf(0) - f'(0)
L {f(n)} (s) = s^£{f}(s) -sn-¹f(0)-sn-²₁'(0)--(n-1)(0)
£¯¹ {F₁+F₂} = £¯¹ {F₁} + £¯¹ {F2}
L-¹{cF} = CL-¹ {F}
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