School

California State University, Long Beach **We aren't endorsed by this school

Course

MATH 2045

Subject

Mathematics

Date

Nov 13, 2023

Type

Other

Pages

3

Uploaded by MasterAnteater1610 on coursehero.com

Math 2045 Exam 5 Prep Worksheet
Chapter 11. 1, 11.2, 10.5, 11.4
Topic
Examples:
1.Classify a differential equation.
Separable DE: the DE can be
written as
q
(
y
)
dy
=
p
(
x
)
dx
Linear first order DE: the DE can
be written as
dy
dx
+
P
(
x
)
y
=
Q
(
x
)
Linear
Separable
a)
dy
dx
=
3
x
+
1
y
b)
dy
dx
=
y
+
x
c)
y
'
=
e
x
(
y
−
1
)
d)
y
'
=
√
y
e)
y
'
=
x
2
−
y
2
2.Write DE for application problems
a) Exponential growth/decay model:
the population is growing/decreasing
at a rate proportional to the population
dy
dt
=
k ∙ y
b) Limited growth model: the
population is growing at a rate
proportional to the difference between
the maximum
N
and the population
dy
dt
=
k
(
N
−
y
)
Logistic growth model: the population
is growing at a rate proportional to the
product of the population and the
fractional difference between the
maximum
N
and the population
dy
dt
=
k ∙ y
(
1
−
y
N
)
Example:
a) A new cell phone is introduced into the market. It is
predicted that sales will grow logistically. The manufacturer
estimates that they can sell a maximum of 70 thousand cell
phones. After 29 thousand cell phones have been sold, sales
are increasing by 4 thousand phones per month. Find the
differential equation describing the cell phone sales,
where
y
=
y
(
t
)
is the number of cell phones (in
thousands) sold after
t
months.
b) The amount of a tracer dye injected into the bloodstream
decreases exponentially, with a decay constant 3% per
minute.
Write the differential equation describing the
situation, where y is the amount of the tracer dye after t
minutes.
c) An isolated fish population is limited to 4000 by the
amount of food available.
If there are now 320 fish and the
population is growing with a growth constant of 2% a year,
write the differential equation to describe the situation,
where y is the fish population after t years.
3.Given solution to a differential
equation, find the unknown parameter
Find the values of
k
for which the function y
(
x
)
=
e
kx
−
2
is a solution of the differential equation
y
'
'
−
8
=
4
y

4.Solve differential equations by
separation of variables.
Find the general solution
dy
dx
=
y
2
5
√
x
5.Solve linear first-order differential
equations by using the integrating
factor.
The amount a full-time student is educated
(
x
)
changes
with respect to the student's age
t
according to the
differential equation
dx
dt
=
1
−
kx
, where k is a constant.
Suppose
x
(
0
)
=
x
0
.
6. Perform matrix operations
2
ae
−
3
t
[
1
−
5
]
+
bt
[
−
2
6
]
7.Find eigenvalues and eigenvectors for a
2x2 matrix
[
−
255
1152
−
56
253
]

8.Write systems of linear differential
equations for compartment problems
in matrix form
9.Solve a system of linear differential
equations using eigenvalues and
eigenvectors