The University of Texas at San Antonio (UTSA)
EGR 2323 Applied Engineering Analysis I
Lesson Study: 1st-Order Linear Modeling
Student:
Write name and UTSA ID (abc123):
Name:
UTSA ID:
Section:
(See options at right)
• Ramirez James
• drh657
0D1
(0A1) MWF 9:00 am
(0B1) MW 10:00 am
(0C1) MWF 2:00 pm
(0D1) MW 4:00 pm
(0E1) MWF 12:00 pm
Teaching Assistant
James Smith
Instructions
In this activity we will study how to formulate and solve differential equations from statements (word
problems).
Students will sort a set of cards that contain four different
word problems. The specific objective is to match up four
cards per problem:
1. The card that contains the proportional relationship
described in the word problem.
2. The card that contains the complete linear model,
which includes the ordinary differential equation and
the initial conditions described in the word problem.
3. The card that contains the general solution and the
particular solution of the word problem.
4. The card that contains the plot of the particular
solution of the word problem.
Cards are matched up by placing them on the spaces
shown in the next slide.
(1)
(2)
(3)
(4)
Set of Cards to Match
Conclusion (Part I)
Students are requested to describe the procedures that they followed while matching the different cards. Think on the
set of steps that were conducted to complete each problem:
Word Problem #1
• In order to find the proportionality relationship, I had to discern the proportional word statement directly into a math
function. In this case since time (t) was directly proportional to the population of ants (y) so its just y. To find the linear
model, I used the initial condition to narrow down the choices then verified using the ODE. Now to find the general and
particular solution I once more used the initial condition to narrow down the choices and verified by solving for the
constant of integration “C”. And Lastly, I matched the graph by verifying the points on the chart with the given conditions.
Word Problem #2
• In order to find the proportionality relationship, I had to discern the proportional word statement directly into a math
function. In this case since time (t) was proportional to the square root of the water level (y) so its √𝑦𝑦. To find the linear
model, I used the initial condition to narrow down the choices then verified using the ODE. Now to find the general and
particular solution I once more used the initial condition to narrow down the choices and verified by solving for the
constant of integration “C”. And Lastly, I matched the graph by verifying the points on the chart with the given conditions.
Word Problem #3
• In order to find the proportionality relationship, I had to discern the proportional word statement directly into a math
function. In this case since time (t) was proportional to one minus velocity (y) so its (1 - y). To find the linear model, I used
the initial condition to narrow down the choices then verified using the ODE. Now to find the general and particular solution
I once more used the initial condition to narrow down the choices and verified by solving for the constant of integration “C”.
And Lastly, I matched the graph by verifying the points on the chart with the given conditions.
Word Problem #4
• In order to find the proportionality relationship, I had to discern the proportional word statement directly into a math function. In this
case since time (t) was proportional to the difference between an object’s temperature (y) and the ambient temperature (𝑦𝑦𝑎𝑎𝑎𝑎𝑎𝑎 ) so its
(y - 𝑦𝑦𝑎𝑎𝑎𝑎𝑎𝑎 ). To find the linear model, I used the initial condition to narrow down the choices then verified using the ODE. Now to find
the general and particular solution I once more used the initial condition to narrow down the choices and verified by solving for the
constant of integration “C”. And Lastly, I matched the graph by verifying the points on the chart with the given conditions.
Conclusion (Part II)
Students are requested to propose a word problem and the corresponding proportional relationship, linear model,
general solution, and particular solution. The word problem does not necessarily have to model reality, but students are
encouraged to study real-world examples.
Word Problem
The rate of change of the
number of polar bears (P)
is directly proportional to a
given time (t). The initial
population of polar bears is
150. After 4 years it is
recorded that there are
only 50 polar bears left.
Proportional Relationship
𝑑𝑑𝑃𝑃
𝑑𝑑𝑡𝑡
𝛼𝛼 P
Linear Model
𝑑𝑑𝑃𝑃
= 𝑘𝑘𝑘𝑘
𝑑𝑑𝑡𝑡
P (0) = 150
P (4) = 50
General/Particular Solution
P = 𝑐𝑐𝑐𝑐 𝑘𝑘𝑘𝑘
𝑃𝑃 = 150𝑒𝑒 −0.27𝑡𝑡