MATH_2250_001_FALL_2023_LAB7
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University of Utah *
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Course
2250
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
4
Uploaded by tinalove801
University of Utah
Fall 2023
MATH 2250-001 Lab 7 Specification
TA: Devanshi Merchant
Subject to Change; Last Updated: 2023-10-18
The printed version of this document may contain errors or
outdated information. Always refer to the digital version on
Canvas for the most accurate and up-to-date content.
1
Background
In this lab, we will learn about solving systems of differential equations.
2
What to Do
Solve the following problems.
1. Consider the following three collections of functions:
I
:
e
4
x
, xe
4
x
, x
2
e
4
x
II
:
e
x
, e
−
x
cos
(
√
2
x
)
, e
−
x
sin
(
√
2
x
)
III
:
{
e
−
2
x
cos(
x
)
, e
−
2
x
sin(
x
)
, xe
−
2
x
cos(
x
)
, xe
−
2
x
sin(
x
)
}
For each collection of functions, complete the following tasks:
(a) Verify that the collection is linearly independent.
(b) Construct a homogeneous linear ODE whose space of solutions is spanned
by the collection.
(c) Write down the general solution of the ODE you constructed in the previ-
ous part.
2. Consider the following
forcing functions
:
I
:
f
(
x
) = 0
II
:
f
(
x
) = sin(
x
)
1
University of Utah
Fall 2023
III
:
f
(
x
) =
e
2
x
sin(
x
)
Use the
method of undetermined coefficients
to find the general solution of the
nonhomogeneous linear ODE
y
′′
−
2
y
′
+
y
=
f
(
x
)
.
3. Consider the following family of nonhomogeneous linear ODEs:
y
′′
+
ay
′
+
by
=
e
sx
.
Here
a, b
and
s
are parameters. Your task is to solve this family of ODEs; it’s
likely that the method of undetermined coefficients will be useful here!
(a) First solve the homogeneous part of the ODE:
y
′′
+
ay
′
+
by
= 0
.
It’s likely that your answer will involve a case analysis based on the sign
of the
discriminant
a
2
−
4
b
. Make sure that in your final answer the de-
pendencies on
a
and
b
are explicit. Recall that the general solution of the
homogeneous part of the ODE is also called the
complementary solu-
tion
of the original, nonhomogenous ODE.
(b) Next, to find a particular solution of the original ODE, try a function of
the form
y
= (
α
2
x
2
+
α
1
x
+
α
0
)
e
σx
,
where
α
0
, α
1
, α
2
, σ
are the undetermined coefficients. Note that while these
four numbers do not depend on the independent variable
x
, they will likely
depend on the parameters
a, b
and
s
.
(c) Finally, adding the particular solution you found in the previous item to
the complementary solution, write down the general solution of the original
ODE.
3
ChatGPT Regulations
This section is in case you decide to use ChatGPT in this lab worksheet. If you will
not be using ChatGPT you may skip this section.
Devanshi Merchant
2
u1474124@utah.edu
University of Utah
Fall 2023
3.1
ChatGPT Versions
•
You may use either the GPT3.5 (freely available with a ChatGPT account) or
GPT4 (available with a ChatGPT Plus account).
•
Turn off all Custom Instructions before you start a chat. If you don’t, it will
be apparent in the archived version that you didn’t.
3.2
Chat Guidelines
•
Your first message in any given chat must be the following guardrails paragraph;
you may copy and paste it:
Hello. I am working on a differential equations and linear algebra
problem as part of a university class. My instructor has permitted
the use of ChatGPT, but only under specific guidelines to
encourage independent critical thinking. Please assist me by asking
probing questions, encouraging reflection, and providing general
insights about the concepts involved. Do not offer direct hints,
strategies, solutions, or step-by-step guidance. I seek to understand
the underlying principles and want to develop my own approach to
the problem. Your role is to facilitate my learning process without
directly leading me to the answer. Thank you!
•
You may copy and paste parts of this specification document, as well as parts
of the textbook or other sources.
•
You may not ask ChatGPT to write for you the solution for any one of the
problems in complete detail.
3.3
Archiving Chats
•
Once you are done with a chat with ChatGPT, click the "Share chat" icon on
the top righthand corner; see the
documentation
for details. In your chat don’t
include any personal information, and keep your user name hidden when you
are creating a link for anonymity.
•
Next you will use
Wayback Machine
to take a "snapshot" of your chat, see the
documentation
for the "Save Page Now" feature. You do not need an
Internet
Archive
account to do this, but having such an account (which is free) would
provide you with further options.
•
You have to take a snapshot of each one of your relevant chats separately, and
share the links to their archived versions in the form for this problem set.
•
To see an example of the outcome, see the Acknowledgements section in the
course syllabus. Note that the staff did use Custom Instructions in this case.
Devanshi Merchant
3
u1474124@utah.edu
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University of Utah
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4
How to Submit
•
Step 1 of 2:
Submit the form at the following URL:
https://forms.gle/akWMZTFPtZN5QN1w9
.
Your submission on Gradescope will receive a zero if you don’t submit this form
before the deadline.
•
Step 2 of 2:
Submit your work on Gradescope at the following URL:
https://www.gradescope.com/courses/565427/assignments/3155701
,
see the Gradescope
documentation
for instructions. As you upload your work,
please separate it into individual problems. This will expedite the response and
feedback process.
oMake sure that you are submitting your work under the
correct assignment on Gradescope.
5
When to Submit
This lab worksheet is due on October 26, 2023 at 11:59 PM.
Devanshi Merchant
4
u1474124@utah.edu