Kevin Brent Akshaya Annavajhala Professor Gurarie Math 224 Lab 2.3: The Harmonic Oscillator with Modified Damping Abstract: This lab analyzed the effects of various modifications to the damping term of the standard harmonic oscillator model: {y' = v,
tion of deer in a particular wooded area is modeled by the
dP 1 P
_._:_ 1_
dt 10P< 2)
Where P is the population in thousands, time is measured in days.
h day; modify the dif
The behavior of the popula
logistic equation
(a) Suppose that a fraction 04 of th
MW.- limw 4 Wm WM? (1): % gwd Wwf
2%:mwi-bvw 6 FMKMWM '31"
G that +37.
Ordinary Differential Equation
Section 1.9: Integration Factors
We want to solve a rst order linear nonhomogenous differential equation
written in the form:
a;"\
7 +g_<t>y 11(3)
ol _ 1
Ordinary Differential Equations
Section 2.2: The Geometry of Systems
Recall the predator-prey system:
dR
= 2R 1.2RF
dt
dF
= F + 0.9RF
dt
We want to write the system of differential equations in vector notation.
dR
dt
Let P(t) =
dF .
dt
dP 2R 1.2RF
Ordinary Differential Equations
Section 1.8: Linear Equations
W
0 Linear or nonlinear DE' ' 2 e2@ sint 062. (51.7- t
,_ ' . +1 +3 15 Q
. 47 TEE
dP ~ ~ z '
0 Linear or nonlinear DE: E = P2 t 2% ,UW , GU:
d
A rst order linear differential equation has the
(Elementary) Differential Equations
Section 1.1: Modeling via Differential Equations
Def. Ordinary differential equation: equation With (ordinary) derivatives in
W_
it. Ordinary means the solution is a function of one variable.
For example:
@:
a
[of
y(t)
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Quiz7
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For the given linear sy
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1. Using the guessandtest method that the
Name: SOlUJ (N15
Math 224 Exam 3
March 6, 2015
1. Here are two different versions of the model for Paul and Bobs cafes, both of which
suppose that current prots from either café have a positive effect on Pauls prots7
and current prots from either cafe hav
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For the given linear system and an initia
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1. Consider the p
Using the two solutions, you can find the general solution as
Instead of find a solution and go the linear combination later, you can directly start with an arbitrary vector and find the soluiton
The arbitrary constants go in to the entries of the vector
Ordinary Differential Equations
Section 2.4: Additional Analytic Methods for Special
Systems
Basic concepts
Dene vector valued function 57 for variables x and y and vector eld 17 for
the right hand side f and g: The system of differential equations
0) cfw
Ordinary Differential Equations
Section 3.7: The TraceDeterminant Plane
Consider the system
c Find eigenvalues.
#VAKI) A b I: (a.,\);.,\) b(, a A2 #aMM italkc =0
C (M =tAMLA-Az)
Arth: AWk: WW): T 7 )u'/\2= MFR = aka/V: 1)
A2" )\+D:0 4:724)
A9 Ti 'T'Zl
Elementary Differential Equations
Section 3.5: Repeated and zero eigenvalues
Consider the system
dY 3 0 3 0
_ = Y, A = _
dt 0 3 0 3
Find the general solution.
Fimtmegmmws Wx/J
dam-AI): 3% O 3 C-a-xvzm
0 a-A
21".)9-7 3
@3 Far 66W M2." ~3. jrnA 6W (2
AV:
Spring 2017, Math 224, Quiz 1 Name[in printmw
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Show your work. Conduct yourself within the guidelines of the University Academic integrity.
dy 2
1. Locate the bifu
Ordinary Differential Equations
Section 2.1: Modeling Via System of Differential
Equations
Recall the predatorprey system:
dR
~= RF
d1; QR 8.
gzVFnLdfF
o What are the equilibria?
R k 02 ~F~):0 <9
Fez; +k)=0 <2?
cfw_2 w r" s":
a _ 3 -
-O : - .1 f
l V R S
Name: , W3
Math 224 Exam 2
February 18, 2015
1. Consider following two population models
dx d3:
= 2 1.2 = 2 - 1.2
or dt 96 it w m.
dy _ d3! _
a; y + 0.933y, dt y 0.933y
(a) One of the models is a predator/ prey system, and the other models two competing
Math 224 Exam 6
December 5, 2012
1. For ac, y > 07 consider the system
Where a is a real parameter.
!6
(a) Show that the system is Hamiltonian and identify the Hamiltonian function.
W $5:ng £33, ,2 O (b) Sketch the level curves of the system for a = %, 1,
Name: gl Umg
Math 224 Exam 5
November 14, 2012
1. Consider the undamped, forced harmonic oscillator modeled by
d2 .
Eg- + 4y = 2cos(wt).
(a) If w 75 2, nd the general solution to the equation above.
(b) If w =. 3, give the solution to the initial valu
1. Two populations, one of predators and one of prey, are modeled by the system of
differential equations
dcc
dy_ y
EZJLyOE) 2:33;
(a) Which of 33(23) and y(t) represents the predator population and which represents
the prey population? Explain your answe
Chapter 2
Even-Numbered Homework Solutions
2.1
2. Find all equilibrium points for the two systems. Explain the significance of these points in terms of the predator and
prey populations.
System (i):
dx
dy
Set
= 0 and
= 0. We have:
dt
dt
x
0 = 10x 1
20xy
Spring 2017
Syllabus
Math224 Elementary Differential Equations
Instructor Dr. Longhua Zhao Office Yost Hall 240
Office Hours(Tentative)
Class Meetings
Phone 216-368-4838
Zhao
Email [email protected]
TueThu 11:15am12:15am or by appointment. You are alw
Even-Numbered Homework Solutions
Chapter 1
1.1
8. Using the decay-rate parameter you computed in 1.1.7, determine the time since death if:
(a) 88% of the original C-14 is still in the material
ln2
The decay-rate parameter from 1.1.7 is =
. Then use the fo
Ordinary Differential Equations
Section 1.7: Bifurcations
dP
Equations with parameters: a 2 EP. WM @614
:lmmw'
0 We want to use the same model for different populations.
0 The proportionality constant Will change.
0 We may not know exactly What k will be.