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 o
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w
Example: p. 72 #58. What does the Existence and Uniqueness Theorem (
say about the solution given the following initial values:
dy = 99)
M E = y(y 1)(y 3) : fig) (mm/v5 R2
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dy
1. Solve the given initialvalue problem 2y z: 36, y(0) = 10.
dt
if: M M .2
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
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
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):
d
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 w
Name: SOlOH
Math 224 Exam 5
April 17, 2015
d2
(a) Solve the initial value problem $2 + 4y 2 300s 2t, y(0) = y(0) = 0.
Half)
Fromm 520101115 12> w WW5 pm, we Cw M MU" m
1.
Homoms pm: 3"+L|J:o =3 k,cos
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.
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.
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Spring 2017, Math 224, Quiz 1 Name(in print): A
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- 2t 2 + 3 2
1. Frist find the general solution of the diff
Different Equations
Section 1.4: Eulers Method
M
0 Numerical methodlfor solving the initial value problem % = f (t,y),
y (to) = yo-
Sketch a solution by drawing graph tangent to slope eld at each
poi
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Name(PRINT):
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Spring 2017
Too Slow
Just about right
Math 224
Quiz7
Too Fast
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Show your work. Conduct yourself within the guidelines of 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
Ordinary Differential Equations
Section 1.8: Linear Equations
W
0 Linear or nonlinear DE' ' 2 [email protected] sint 062. (51.7- t
,_ ' . +1 +3 15 Q
. 47 TEE
dP ~ ~ z '
0 Linear or nonlinear DE: E = P2 t 2% ,UW ,
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 vecto
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 eq
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
Spring 2017, Math 224, Quiz 1 Name[in printmw
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Show your work. Conduct yourself within the guidelines of the
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;
. ve): M) 9?
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3?:an igtudt
VW'A
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Nah/=0
@290 $359wa ewflym
* 3%: o-gunqs %=b Mixing Problems: TWO or more substances are are mixed together at var
ious rates. Generally want to nd
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 pre
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
Uniqueness & Existence theorem
1
1
Basic theorems
For the I.V.P.
dy
= f (t, y), y(t0 ) = y0 ,
dt
we could establish the following results:
* Existence theorem: Suppose f (t, y) is a continuous functio
Damped harmonic oscillator
1
1
Basic concepts
The damped harmonic oscillator
dy
d2 y
+b
+ ky = 0
2
dt
dt
could be formulated into (with p and q being constants)
(1)
m
d2 y
dy
+p
+ qy = 0.
dt2
dt
(2)
C
Linear equations
1
1
Basic concepts
What is linear first-order D.E.?
dy
dy
dy
= a(t)y + b(t), eg.
= t2 y + cos t, but not
= y2
dt
dt
dt
where b(t) term is the non-homogeneous term, or forcing term.
L
System of equations
1
1
Basic concepts
System of autonomous D.E.s
dx
= f (x, y)
dt
dy
= g(x, y)
dt
For example,
f (x, y) = 3x + y, g(x, y) = x y
f (x, y) = 2x(1 x2 ) xy, g(x, y) = 3y(1 y3 ) 2xy
Impo
Euler Method
1
1
Basic concepts
Explicit Eulers method for I.V.P.
dy
= f (t, y), y(0) = y0 .
dt
1) Given time tn , and the supposed solution yn at the time, compute the slope f (tn , yn ) by using the