value problems for second-order linear differential equations with constant coefficients. Many Sections in this chapter are numerous initial- can be interpreted as models of particular physical systems, but usually we do not point this out explicitly. Problems slogany In each of Problems 1 through 7, find the inverse Laplace transform of the given function. 3 1. F(s) = 2. F(s) = 3. F(s) = 4. F(s) = 5. F(s) = 6. F(s) = 7. F(s) $² = +4 4 85²-4s + 12 s(s² + 4) 1-2s s² + 4s +5 In each of Problems 8 through 16, use the Laplace transform to solve the given initial value problem. (s - 1)³ 2 s² + 35-4 2s+2 s²+2s +5 2s-3 52-4 imonylag y(0) = 1, y'(0) = -1 8. y" - y'-6y=0; 9. y" + 3y' +2y=0; y(0) = 1, y'(0) = 0 (@))2 10. y" - 2y + 2y = 0; 11. y" - 2y + 4y = 0; y(0) = 0, y'(0) = 1 y(0) = 2, 12. y" +2y + 5y = 0; y(0) = 2, 13. y(4) - 4y"" + 6y" - 4y' + y = 0; y'(0) = 1, y"(0) = 0, y""(0) = 1 14. y(4) - y = 0; y(0) = 1, y'(0) = 0, y" (0) = 1, y"" (0) = 0 15. y"+w²y = cos(2t), w² #4; y(0) = 1, y'(0) = 0 da 16. y" - 2y' +2y=et; y(0) = 0, y'(0) = 1 In each of Problems 17 through 19, find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem. A method of determining the inverse transform is developed in Section 6.3. You may wish to refer to Problems 16 through 18 in Section 6.1. 17. y" +4y= [1, 0≤t<,vis ad to motanim non y(0) = 1, y'(0) = 0 π ≤t< 00; 0, ft, 0≤t < 1, 1, 1≤t<∞0; 18. y" + 4y = t, amaldor y'(0) = 0 motanen sosiqa y'(0) = -1 y(0) = 0, 19. y"+y=2-t, 0, y(0) = 0, y'(0) = 0 0 ≤ t < 1, 1st <2, y(0) = 0, y'(0) = 0 2 ≤t <∞0; ad
value problems for second-order linear differential equations with constant coefficients. Many Sections in this chapter are numerous initial- can be interpreted as models of particular physical systems, but usually we do not point this out explicitly. Problems slogany In each of Problems 1 through 7, find the inverse Laplace transform of the given function. 3 1. F(s) = 2. F(s) = 3. F(s) = 4. F(s) = 5. F(s) = 6. F(s) = 7. F(s) $² = +4 4 85²-4s + 12 s(s² + 4) 1-2s s² + 4s +5 In each of Problems 8 through 16, use the Laplace transform to solve the given initial value problem. (s - 1)³ 2 s² + 35-4 2s+2 s²+2s +5 2s-3 52-4 imonylag y(0) = 1, y'(0) = -1 8. y" - y'-6y=0; 9. y" + 3y' +2y=0; y(0) = 1, y'(0) = 0 (@))2 10. y" - 2y + 2y = 0; 11. y" - 2y + 4y = 0; y(0) = 0, y'(0) = 1 y(0) = 2, 12. y" +2y + 5y = 0; y(0) = 2, 13. y(4) - 4y"" + 6y" - 4y' + y = 0; y'(0) = 1, y"(0) = 0, y""(0) = 1 14. y(4) - y = 0; y(0) = 1, y'(0) = 0, y" (0) = 1, y"" (0) = 0 15. y"+w²y = cos(2t), w² #4; y(0) = 1, y'(0) = 0 da 16. y" - 2y' +2y=et; y(0) = 0, y'(0) = 1 In each of Problems 17 through 19, find the Laplace transform Y(s) = L{y} of the solution of the given initial value problem. A method of determining the inverse transform is developed in Section 6.3. You may wish to refer to Problems 16 through 18 in Section 6.1. 17. y" +4y= [1, 0≤t<,vis ad to motanim non y(0) = 1, y'(0) = 0 π ≤t< 00; 0, ft, 0≤t < 1, 1, 1≤t<∞0; 18. y" + 4y = t, amaldor y'(0) = 0 motanen sosiqa y'(0) = -1 y(0) = 0, 19. y"+y=2-t, 0, y(0) = 0, y'(0) = 0 0 ≤ t < 1, 1st <2, y(0) = 0, y'(0) = 0 2 ≤t <∞0; ad
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
15
Transcribed Image Text:this and other sections in this chapter are numerous initial-
value problems for second-order linear differential equations with constant coefficients. Many
can be interpreted as models of particular physical systems, but usually we do not point this
out explicitly.
JS
nibysaisies the graph of
Torgny
In each of Problems 1 through 7, find the inverse Laplace transform
of the given function.
3
$(7)
(1)g
Problems
1. F(s):
=
5² +4
(0)1 Ls (1)83 4
2. F(s) =
3. F(s) =
4. F(s) =
ooo oldsbe
ang 2s +2
5. F(s) =
=
6. F(s) =
7. F(s) =
(s - 1)³
2
s²+3s-4
lang coalgs.1 od
s²+2s +5
2s-3
5²-4
8. y" - y' - 6y=0;
9. y" + 3y' + 2y = 0;
(2) bus (2)D II
vlovinggren
Inicionylag a
mony
Howeve
y(0) = 1, y'(0) = -1
y(0) = 1, y'(0) = 0
10.
11.
8s² - 4s +12
s(s² + 4)
1-2s
s² + 4s +5
In each of Problems 8 through 16, use the Laplace transform to solve 18. y" + 4y =
the given initial value problem.
for
1
0 <
y" - 2y' + 2y = 0;
y" - 2y + 4y = 0;
tai odi 16 noilonul locas a
(5)
17. y" + 4y =
siqu
12. y" +2y' + 5y = 0; y(0) = 2,
13. y(4) - 4y"" + 6y" - 4y' + y = 0;
y'(0) = 1, y"(0) = 0, y""(0) = 1
14. y(4) — y = 0;_y(0) = 1, y'(0)
y" (0) = 0
(166
19. y"+y=2-1,
0,
92
(0
y(0) = 0, y'(0) = 1
y(0) = 2,
y'(0) = 0
y'(0) = -1
y(0) = 0,
od indignimuezA
viher,mai-yd-oriol
f1, 0≤t<T,
0, π ≤ t < ∞0;
0 ≤ t < 1,
1, 1≤t < 0;
y(0) = 1, y'(0) = 0
15. y"+w²y = cos(2t), w² #4;
16. y" - 2y' + 2y = -¹; y(0) = 0, y'(0) = 1
In each of Problems 17 through 19, find the Laplace transform Y(s) =
L{y} of the solution of the given initial value problem. A method of
determining the inverse transform is developed in Section 6.3. You
may wish to refer to Problems 16 through 18 in Section 6.1.
bas
I am
amaldor
anem sosiqa
= 0, y" (0) = 1,
13 II
y(0) = 1, y'(0) = 0
st=(1)
y(0) = 0, y'(0) = 0
sd
ban
SC
0 ≤ t < 1,
an A
1≤t<2, y(0) = 0, y'(0) = 0
2≤t < 8;
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