NetMath at the University of Illinois

Math 285 DiffEq&Mathematica
Courseware Syllabus

Part I: Transition from Calculus: Classical Theory of Differential equations

DE.01  Transition from calculus: The Exponential Diffeq   y'[t] + r y[t] = f[t]

How to write down formulas for solutions of y'[t] + r y[t] = 0.
How to use integrating factors to get formulas for solutions of y'[t] + r y[t] = f[t].
If r > 0, then all solutions of y'[t] + r y[t] = f[t] go into the same steady state.
Exponential models.
The jump function UnitStep[t-d] and the impulse function DiracDelta[t-d]
Impulse forcing the exponential diffeq with a Dirac Delta function; The physical meaning of the impulse force.
The superpostion principle.

DE.02  Transition from Calculus : The Forced Oscillator diffeq   y''[t] + b y'[t] + c y[t] = f[t]

DE.03  Transition from Calculus : Laplace Transform and Fourier Analysis

Part 2: Introduction to Modern Theory of Differential Equations

DE.04  Modern Diff Eq Issues

DE.05 Modern Diff Eq: First Order Differential Equations

Reading an autonomous diffeq through phase lines
Automomous diffeqs with parameters. Bifurcations and bifurcation points
Hand symbol manipulation: Separating the variables
Population models and control
Using bifurcation plots to study E. Coli growing in a chemostat
Automatically controlled air conditioning
Getting there in infinite time versus getting there in finite time

DE.06 Modern Diff Eq: Systems and Flows

Flows and their trajectories as pairs of solutions of a system of differential equations
Flow analysis of  the unforced linear oscillator differential equation by converting it to a system of two first order differential equations
Equilibrium points
Damped oscillators, undamped oscillators and van der Pol's nonlinear oscillator
Linear systems and graphical meaning of eigenvectors of the coefficient matrix
Pursuit models
Boundary value problems: Shooting for a specified outcome

DE.07  Modern DiffEq: Eigenvectors and Eigenvalues for Linear Systems

Eigenvectors of the coefficient matrix point in the directions of strongest inward and/or outward flow
Eigenvalues of the coefficient matrix indicate realtive strenghs of inward and/or outward flow
Eigenvalue-trajectory analysis to predict swirl in,swirl out or no swirl at all
Stability and instability
Reservoir Models for drug metabolization
Linear systems in life science, chemistry and electrical engineering
Higher dimensional linear systems

DE.08  Modern DiffEq: Linearization of Nonlinear Systems

Using the Jacobian to approximate a nonlinear diffeq system by linearizing at equilibrium points
Attractors and repellers: Lyapunov's rules for detecting them via analysis of the eigenvalues of the Jacobian
The pendulum oscillator: Damped and undamped
When linearizations can be trusted and when they shouldn't be trusted  
Linearization of pendulum oscillators: Using linearzation to estimate the amplitude and frequency of a pendulum oscillator
Energy and the undamped pendulum oscillator
The Van der Pol oscillator
Gradient and Hamiltonian systems
Lorenz's chaotic oscillator

Part 3:  Partial DiffEq: Heat and Wave Equations

DE.09 Heat Equation and Wave equation  

Rigging f[t] on [0, L] to get a pure sine fast Fourier fit of f[t] on [0, L]
Fourier Sine fit for solving the heat equation  
Fourier Sine fit for solving the wave equation
Solving the heat and the wave equations in the case that initial data are given by a data list

Created by Mathematica  (September 6, 2006)

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