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Contents
Helper Application Tutorial
Numerical Solutions of Differential Equations
World Class Sprints
Logistic Growth Model
Predator-Prey Models
Second-Order Linear Homogeneous Differential Equations
with Constant Coefficients
Spring Motion
Forced Spring Systems I
Matrix Operations
Eigenvalues and Eigenvectors
Trajectories of Linear Systems
The Pendulum
Lead in the Body
Gain and Phase Shift
The van der Pol System
Helper Application Tutorial
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Purpose:
To learn the basics of Maple V, Release 4 or Release 5
for use in differential equations modules.
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Prerequisites:
None
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Available for:
Maple
Numerical Solutions of Differential Equations
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Purpose:
To gain experience with numerical methods for approximating
the solution to
first-order initial value problems.
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Prerequisites:
Work through the basic tutorial for your computer algebra system.
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Available for:
Maple
World Class Sprints
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Purpose:
To explore the applicability of a linear differential
equation as a model for the
process of sprinting, and to illustrate the importance
of parameters in modeling.
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Prerequisites:
The tutorial for your helper application and ability to solve
a first-order linear
differential equation with constant coefficients.
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Available for:
Maple
Logistic Growth Model
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Purpose:
To study a standard model of population growth in a
constrained environment.
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Prerequisites:
Separation of variables.
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Available for:
Maple
Predator-Prey Models
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Purpose:
To develop and explore the Lotka-Volterra model
for predator-prey interactions as a prototypical first--order
system of differential equations.
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Prerequisites:
The module on Numerical Solutions of
differential equations.
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Available for:
Maple
Second-Order Linear Homogeneous Differential Equations with Constant Coefficients
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Purpose:
To investigate the qualitative behavior of the
solutions of initial value problems of the form
y" + ay' + by = 0,
y(0) = y0,
y'(0) = y1.
In particular, to determine how solutions depend on the
signs and magnitudes of the coefficients a and b and on the
initial conditions.
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Prerequisites:
The helper application tutorial and
knowledge of the symbolic form of solutions of differential
equations of the form y" + ay' + by = 0.
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Available for:
Maple
Spring Motion
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Purpose:
To investigate the
mathematical model
y'' + (c/m)y' + (K/m)y = 0
for spring motion and to study the effect
of increased damping.
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Prerequisites:
Knowledge of second-order linear homogeneous
differential equations with constant
coefficients.
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Available for:
Maple
Forced Spring Systems I
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Purpose:
To explore the effects of an external driving force
on a simple linear oscillator, damped or undamped.
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Prerequisites:
The Spring Motion module and knowledge
of the symbolic form of solutions of differential
equations of the form y" + ay' + by = f(t),
where f is a sine or cosine function.
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Available for:
Maple
Matrix Operations
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Purpose:
To experiment with
matrix operations, espcially
multiplication, inversion, and
determinants, and to explore
applications to solving systems of
linear equations. In the process of
studying these matrix operations, we
will learn how to use a helper
application to carry out matrix
computations.
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Prerequisites:
A basic understanding of linear combinations of vectors,
familiarity with matrix multiplication.
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Available for:
Maple
Eigenvalues and Eigenvectors
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Purpose:
To experiment with and explore properties of
eigenvalues and eigenvectors and their application to
differential equations.
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Prerequisites:
The Matrix Operations module and the
concept of reduced row echelon form.
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Available for:
Maple
Trajectories of Linear Systems
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Purpose:
To investigate the trajectories in the phase plane
of 2x2 homogeneous linear systems of first-order
differential equations of the form X' = AX.
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Prerequisites:
The Matrix Operations module and an
understanding of the meaning of eigenvalues and
eigenvectors of the matrix A.
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Available for:
Maple
The Pendulum
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Purpose:
To explore the phase plane for a second-order
nonlinear differential equation, specifically the standard
model for damped and undamped pendulums.
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Prerequisites:
The Spring Motion module.
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Available for:
Maple
Lead in the Body
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Purpose:
To develop and explore a compartment model
for the amount of lead in a human body and to study a
three-dimensional driven linear system.
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Prerequisites:
The module on Trajectories of Linear
Equations.
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Available for:
Maple
Gain and Phase Shift
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Purpose:
To explore the relationship between frequency of
an external driving force and the parameters of a damped
linear oscillator.
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Prerequisites:
The module on Forced Spring Motion.
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Available for:
Maple
The van der Pol System
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Purpose:
To explore the van der Pol model for a nonlinear
electrical circuit -- in particular, to study the limit cycle
phenomenon.
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Prerequisites:
The module on Forced Spring Motion.
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Available for:
Maple
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