Q6 Find Laplace transforms of the following functions: if 0
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
In part a, since the question was being added 6 and subtracted 6, where did the minus 6 go in the answer? Can you explain the process?
![Q6 Find Laplace transforms of the following functions:
Recall: L[u₂lt) · f(t-c)) = ē"F(s)
e
(a) f(t) =
cos 3t if 0 < t < 2
if 2 <t
4
You may find the identity cos(x + y) = cos x cos y — sin æ sin Y useful.
f(t) = cos 3€ (1-u₂lt)) + 4u₂lt)
=
cos 3t - 4₁₂14) cos(3 (t-2) + 6) + 4×₁₂ (4)
Need £ (cos(3t+6)):
cos (3++6)= (cos 6) cos 3€ - (5 in 6) sin 3€
So L (cos(34+6)) =
(cos6) 5.
3 sinb
s²+9
3²+9
=
F(₁): 33+9 - 2²¹ (2016) - 35-4) + Ye"
sin
-23 /
e
+9/
$
(b) f(t) = [t − 2] [Hint: write as a piecewise function!]
(-(1-2) 0≤t<2
f(t) =
[+-2
+22
= -(t-2) (1 − u₂lt)) + (+-2) u₂lt)
= 2- + +
2(t-2) 4₂ (+)
- 2s
F(s) = = = = = = + 2 = = =
2
S
mimizživinſke kanalväržátko nákuvažnizimininkamusta kasvava miniszwiesiskai įázávažímmužůní](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae98b63a-c1e0-40a0-8ef8-c296ebcafc68%2Fd7627113-dd3b-488c-95a0-d1631c5088b1%2F7nkkg5s_processed.png&w=3840&q=75)
Transcribed Image Text:Q6 Find Laplace transforms of the following functions:
Recall: L[u₂lt) · f(t-c)) = ē"F(s)
e
(a) f(t) =
cos 3t if 0 < t < 2
if 2 <t
4
You may find the identity cos(x + y) = cos x cos y — sin æ sin Y useful.
f(t) = cos 3€ (1-u₂lt)) + 4u₂lt)
=
cos 3t - 4₁₂14) cos(3 (t-2) + 6) + 4×₁₂ (4)
Need £ (cos(3t+6)):
cos (3++6)= (cos 6) cos 3€ - (5 in 6) sin 3€
So L (cos(34+6)) =
(cos6) 5.
3 sinb
s²+9
3²+9
=
F(₁): 33+9 - 2²¹ (2016) - 35-4) + Ye"
sin
-23 /
e
+9/
$
(b) f(t) = [t − 2] [Hint: write as a piecewise function!]
(-(1-2) 0≤t<2
f(t) =
[+-2
+22
= -(t-2) (1 − u₂lt)) + (+-2) u₂lt)
= 2- + +
2(t-2) 4₂ (+)
- 2s
F(s) = = = = = = + 2 = = =
2
S
mimizživinſke kanalväržátko nákuvažnizimininkamusta kasvava miniszwiesiskai įázávažímmužůní
![3:11 PM Tue Jun 7
<
...
Practice final exam solutions.pdf
Q6 Find Laplace transforms of the following functions:
Recall:
cos 3t if 0< t < 2
(a) f(t)
if 2 ≤t
You may find the identity cos(x + y) = cos x cos y sin æ sin y useful.
neily menti
f(t) = cos 3€ (1-u₂lt)) + 4u₂ (t)
=
cos 3t - u₂14) cos 3 (t-2); 6) + 4u, (+)
Need £ (cos(3t+6))
So L (cos(36+6)) = (050) 5
s²+9
F(s)= 335 3249
S
₂-2₁ / (C056)s
(cost)s
5249
е
3 sin6). +
5219
(b) f(t)|t2| [Hint: write as a piecewise function!]
(-(1-2) ost²2
f(t) =
| +-2
+22
3
= -(t-2) (1 -u₂lt)) + (+-2) u₂lt)
2-+ + 2(t-2) u₂(t)
-2s
2e
1
= L(u₂lt) · f(t-c)) = @" "F(s)
cos (3++6)= (cod 6) cos 3t - ( in 6) sin 3t
3 sinb
$²+9
F(s) = = = = = /2/201
4e-²s
S
abrikant zitvichwa ki
29% 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae98b63a-c1e0-40a0-8ef8-c296ebcafc68%2Fd7627113-dd3b-488c-95a0-d1631c5088b1%2Fj3pbdn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3:11 PM Tue Jun 7
<
...
Practice final exam solutions.pdf
Q6 Find Laplace transforms of the following functions:
Recall:
cos 3t if 0< t < 2
(a) f(t)
if 2 ≤t
You may find the identity cos(x + y) = cos x cos y sin æ sin y useful.
neily menti
f(t) = cos 3€ (1-u₂lt)) + 4u₂ (t)
=
cos 3t - u₂14) cos 3 (t-2); 6) + 4u, (+)
Need £ (cos(3t+6))
So L (cos(36+6)) = (050) 5
s²+9
F(s)= 335 3249
S
₂-2₁ / (C056)s
(cost)s
5249
е
3 sin6). +
5219
(b) f(t)|t2| [Hint: write as a piecewise function!]
(-(1-2) ost²2
f(t) =
| +-2
+22
3
= -(t-2) (1 -u₂lt)) + (+-2) u₂lt)
2-+ + 2(t-2) u₂(t)
-2s
2e
1
= L(u₂lt) · f(t-c)) = @" "F(s)
cos (3++6)= (cod 6) cos 3t - ( in 6) sin 3t
3 sinb
$²+9
F(s) = = = = = /2/201
4e-²s
S
abrikant zitvichwa ki
29% 0
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Yeah i get that the question added and subtracted 6, but i was wondering why in the answer it did not answer the subtract 6 in the answer, they only used the add 6 to fit in the equation.
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