APPM 2360 Differential Equations Lab 2: Harmonic Oscillator with Modified Damping
Michael Blanchard 810-07-6309 Professor: David Bortz T/A: Jason Hammond 10AM 023
Ben Maples 810-13-1335 Professor: Will Heuett T/A: Will Heuett 2PM 035
Aaron Willoug

APPM 2360 Lab 1 Fish Population Modeling
Spring 2008
Filip Maksimovic (810689087) Theodoros Horikis Recitation 34: John Vilavert Michaela Cui (810585604) Theodoros Horikis Recitation 036: Sean Nixon
Brandon Parks (810689666) Theodoros Horikis Reci

CHAPTER
7
Nonlinear Systems of Differential Equations
7.1
! 1.
Nonlinear Systems
Review of Classifications x = x + ty y = 2 x + y + sin t Dependent variables: x, y Parameter: Nonautonomous linear system Nonhomogeneous ( sin t ) 2. u = 3u + 4

APPM 2360
12 December, 2013
Week #15
1. (a) If A is a symmetric matrix such that A5 = 0, then A = 0.
(b) If A and B are 3 3 symmetric matrices, then AB is symmetric.
(c) If a 3 3 matrix A is invertible, then its rows form a basis.
(d) The rank of a 4 3 ma

Final Exam Review: Chapters 1-3
1. Question # 1 Direction Fields [1, p.10]
For each of the dierential equations
i) Draw the direction eld
ii) Draw the isolclines for c = 0, c = 1, and c = 2
iii) Describe the long term behavior of solutions.
a) y = 2 + t y

APPM 2360
12 December, 2013
Final Review: Second Part
1. (a) Consider the matrix A:
1/2 1/2
1/2 1/2
Find the eigenvalues of A.
(b) Is 0 a stable equilibrium point for the linear system
dx
= Ax
dt
(c) Describe,how the solution curves of
dx
dt
= Ax look lik

APPM2360: Exam 1 Review Problems
1. True/False (Mixed Concepts)
(a) The ODE 2ty 3t = 0 is homogeneous.
(b) The Runge Kutta Method of order 4 is more accurate than Eulers Method.
(c) The ODE y y admits a semi-stable equilibrium solution at y = 0.
(d) Every

APPM 2360
APPM 2360
25 April, 2013
November 14th, 2013
Week #15
Week #12
1. Consider the matrix
4 3
21
A=
(a) Find eigenvalues and eigenvectors of A
(b) Find the general solution to the system.
2. Consider the system of dierential equations
x = x + y,
y =

APPM 2360
5 December, 2013
Week #14
1. Consider the system
x = 20 x2 y 2
y = 8 xy
(a) What are equilibrium points of this system?
(b) Classify each equilibrium point for this system.
2. Consider matrix and the vector
1 0 3 1
A = 0 1 2 1 ,
1 1 1 1
1
b= 1

APPM 2360
5 December, 2013
Week #14
1. Consider the system
x = 20 x2 y 2
y = 8 xy
(a) What are equilibrium points of this system?
(b) Classify each equilibrium point for this system.
Solution:
(a) For equilibrium points we have to solve system
x2 + y 2 =

APPM 2360
12 December, 2013
Week #15
1. (a) If A is a symmetric matrix such that A5 = 0, then A = 0.
(b) If A and B are 3 3 symmetric matrices, then AB is symmetric.
(c) If a 3 3 matrix A is invertible, then its rows form a basis.
(d) The rank of a 4 3 ma

APPM 2360: Exam I Review Questions
1. Solve the ODE:
t3 y = ln t 2t2 y
Solution: We can rewrite the ODE as
2
ln t
y + y= 3
t
t
By method of integrating factor, we have
(t) = e
2
dt
t
= e2 ln t = t2
thus,
d 2
ln t
2
ty =
t2 y + y =
t
dt
t
Integrating by pa

8
Slope Fields: Graphing Solutions
Without the Solutions
Up to now, our efforts have been directed mainly towards nding formulas or equations describing
solutions to given differential equations. Then, sometimes, we sketched the graphs of these
solutions

ROOTS OF POLYNOMIALS
Assume that we have normalized the polynomial so that the leading coefficient is equal to one:
p n (x) = x n + a 1 x n1 + a 2 x n2 + . + a n1 x + a n
Roots; explicit formulas:
(1)
Allowing for multiple roots and for complex roots, pn(

APPM 2360
Lab #1: Pollution
1
Instructions
Labs may be done in groups of 3 or less. One report must be turned in for each group and must
be in PDF format. Labs must include each students:
Name
Student number
Recitation number
This lab is due on Friday,

APPM 2360: Section exam 1
September 21, 2011.
ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructors name, (3)
your recitation section number and (4) a grading table. Text books, class notes, and calculators
are NOT permitted. A one page

Solution: APPM 2360
Exam1
Spring 2013
Problem 1: (15 points)
Suppose that
dy
= (t 1) |y |,
dt
y (0) = y0 ,
where y0 is a real constant.
(a) [5] Consider y0 = 1. What does Picards Theorem allow you to conclude about the
existence and/or uniqueness of a sol

APPM 2360
30 January, 2014
Week #3
1. Suppose L is a linear operator and y (t), z (t), and w(t) are functions such that L[ y (t) ] = 3et ,
L[ z (t) ] = 0, and L[ w(t) ] = 2et .
(a) What is L[ 3y (t) ]?
(b) What is L[ y (t) + 2z (t) ]?
(c) What is L[ 3y (t

APPM 2360
13 January, 2014
Week #1
t
2
1. Verify that y = et
2
2
es ds + et is a solution of the dierential equation y 2ty = 1.
0
2. Sketch the direction eld for y = y t. What can you say about the long term behavior of
the solutions?
3. Match the dierent

APPM 2360
13 January, 2014
Week #1
t
2
1. Verify that y = et
2
2
es ds + et is a solution of the dierential equation y 2ty = 1.
0
Solution:
t
2
y = et
2
2
es ds + et
0
y
t
d t2
e
dt
2d
= et
dt
2
=e
t2
0
t
2
es ds + et
=
2
es ds +
0
t2
e
+ 2te
d t2
e
dt
t

APPM 2360
23 January, 2014
Week #2
1. Consider the dierential equation, y =
2y
t.
(a) State Picards existence and uniqueness theorem regarding solution(s) to the IVP: y =
f (t, y ), y (t0 ) = y0 .
(b) What does Picards Theorem tell us about the solutions

APPM 2360
November 7th, 2013
Week #11
1. Consider an unforced mass-spring system with mass m = 1/2, spring constant k = 3and
damping constant b = 3 p, for constant p > 0.
(i) Write down the dierential equation which describes the motion of the spring.
(ii

APPM 2360
November 7th, 2013
Week #11
1. Consider an unforced mass-spring system with mass m = 1/2, spring constant k = 3and
damping constant b = 3 p, for constant p > 0.
(i) Write down the dierential equation which describes the motion of the spring.
(ii

APPM 2360 Summer 2017
Lab #1: Fish Population
1
Instructions
Labs may be done in groups of 2 or 3 (i.e., not alone). You may use any programming language
you wish but MATLAB is highly suggested. One report must be turned in for each group and must
be in P