Lab7 Answer Template(6)
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Drexel University, College of Engineering
2022-2023 Academic Year
Drexel University
Office of the Dean of the College of Engineering
ENGR 232 – Dynamic Engineering Systems
Section : 61
Name: _ Stanley _ _ _ _ _ _ _ _ Zhu _ _ _ _
first
last
Lab 7 Answer Template: Laplace Workshop: Summer 2023 Part A: Definition of the Laplace Transform.
Grading: TA will randomly pick one part from each of questions 1 – 5 and award 1 point if correct. Question 1:
a.
f
(
t
)
=
t
b.
f
(
t
)
=
t
2
c.
f
(
t
)
=
3
e
5
t
Record answers in the boxes above.
Question 2:
Find each of the following Laplace transforms. 2a.
f
(
t
)
=
t
n
2
b.
f
(
t
)
=
sin
(
at
)
2
c.
f
(
t
)
=
e
at
Record answers in the boxes above. Also, include any assumptions you needed to present the answer in the "clean" form seen in the Tables. Question 3:
Find these transforms using laplace()
. 3a.
f
(
t
)
=
3cosh 5
t
3
b.
f
(
t
)
=(
t
−
3
)
2
∙u
(
t
−
3
)
3
c.
f
(
t
)
=
√
t
Record answers in the boxes above.
Question 4:
Find the inverse
Laplace transform for each of the following functions defined in the s
-domain. 4a.
F
(
s
)=
1
4
b.
F
(
s
)
=
5
s
+
8
s
2
+
16
4
c.
F
(
s
)
=
1
s
3
/
2
Record answers in the boxes above.
L
{
t
}=
¿
1/s^2
L
{
t
2
}=
¿
2/s^3
L
{
3
e
5
t
}=
¿
3/s-5
L
{
t
n
}=
¿
gamma(n + 1)/s^(n + 1)
L
{
sin
(
at
)}=
¿
gamma(n + 1)/s^(n + 1)
L
{
e
at
}=
¿
-1/(a - s)
L
{
3cosh5
t
}=
¿
(3*s)/(s^2 - 25)
L
{
(
t
−
3
)
2
∙u
(
t
−
3
)
}=
¿
L
{
√
t
}=
¿
f
(
t
)=
¿
f
(
t
)=
¿
f
(
t
)=
¿
1
Drexel University, College of Engineering
2022-2023 Academic Year
Part B: Partial fraction expansions. Question 5:
Find the partial fraction expansion for each of the following functions in the s
-domain. Use partfrac()
.
5a. F
(
s
)
=
16
s
2
−
8
s
5
b.
F
(
s
)
=
9
s
2
−
52
s
+
72
(
s
−
2
) (
s
−
3
) (
s
−
4
)
5
c.
F
(
s
)
=
3
s
2
−
14
s
+
20
(
s
−
3
)
3
Grading: TA will randomly pick one part from each of questions 1 – 5 above and award 1 point if correct. Part C: Solving a Differential Equation using the Laplace Transform. Question 6: DE
:
y
' '
+
y
=
6sin 2
t
and initial conditions: IC
:
y
(
0
)=
0
, y'
(
0
)=
6
Question 7:
Record both
the solution Y
(
s
)
in partial fraction form
and the solution y
(
t
)
in the time-domain that were just found using the Laplace technique here. Did you get the same answer for y
(
t
)
?
Part D: Solve a new DE using the Laplace transform technique. (3 points)
The last three points will be earned by using code similar to that given above to solve the new differential equation: DE
:
y
' '
+
y
'
+
5
4
y
=
13
∙e
−
2
t
IC
:
y
(
0
)
=
4
, y '
(
0
)=
2
Points 8-10: Solve this new DE using the Laplace technique and past these three
answers below. Question 6: Record the exact solution for y
(
t
)
found using dsolve:
y
(
t
)=
¿
¿
Questions 8-10:
8: Y
(
s
)
=
¿
Must be a quadratic over a cubic forpoints. 9: Y
(
s
)
asa
∂
fraction
=
¿
10: y
(
t
)
=
¿
Question 7:
Y
(
s
)
=
¿
(must be in partial fraction form)
y
(
t
)
=
¿
2
Drexel University, College of Engineering
2022-2023 Academic Year
Ready to Submit? Be sure all ten questions are answered. When your lab is complete, be sure to submit three files:
1.
Your completed Answer Template
as a PDF file 2.
A copy of your MATLAB Live Script
3.
A PDF
copy of your MATLAB Live Script
(Save-Export to PDF…)
The due date is the day after your lab section by 11:59pm
to receive full credit. You have one more day, to submit the lab (but
with a small penalty), and then the window closes for good and your grade will be zero.
3
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