Syllabus:
Differential Equations&Mathematica
Authors:  Bill Davis and Jerry Uhl  ©1999
Producer:  Bruce Carpenter
Publisher:         Distributor:  

These lessons are used in Math 385, 286 and 351.

Part 1:  Transition from Calculus: Classical Theory of Differential Equations

DE.01  Transition from Calculus: The Exponential DiffEq: 
           

How to write down formulas for solutions of .
How to use integrating factors to get formulas for solutions of .
If , then all solutions of  go into the same steady state.
Exponential models.
The jump function  and the impulse function 
Impulse forcing the exponential diffeq with a Dirac Delta function; The physical meaning of the impulse force.
The superpositio n principle.

DE.02  Transition from Calculus : The Forced Oscillator DiffEq:  

The undamped unforced oscillator < br>The damped unforced oscillator  
The damped forced oscillator < br>Steady state and transients for forced damped oscillators
Resonance and beating
Euler Identity
The characteristic equation
Using  convolution i ntegrals to try to get formulas solutions  of the forced 
  oscillator diffeq  
Resonance
Forcing an oscillator with a Dirac Delt a function; 
the physical meaning of the impulse hit
Amplitude and frequency of unforced oscillators
Underdamped,critically damped,and overdamped oscillatorsBoundary value problems

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

The Laplace transform of a function 
How to write down the Laplace tranform of the solution of a forced oscillator 
  diffeq
Solving forced oscillator diffeqs by inverting Laplace tran sforms
Fast Fourier point fit and Fourier integral fit 
Combining Fourier fit and the Laplace transform to come up with good 
  approximate formulas for periodically forced oscillators
Fourier analysis for detecting resonance

DE.04 Modern DiffEq Issues
        

Euler's method of faking the plot of the solution of a differential equation 
  and how it highlights the fundamental issue of diffeq
Reading a diffeq t hrough  flow plots
Solving diffeq's numerically with Mathematica
Systems of interacting differential equations: The predator-prey model
Sensitive dep endence on starter data 
The drinking versus driving model
Population models and control; Logistic harvesting
Lanchester war model

DE.05 Modern DiffEq: 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 t here in infinite time versus getting there in finite time

DE.06 Modern DiffEq: 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 equa tion by converting it to a system of two first order differential equations
Equilibrium points
Damped oscillators, undamped oscillators and van der Pol's nonlinea r 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 streng hs 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: Linearizations 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 ana lysis of the eigenvalues of the Jacobian
The pendulum oscillator: Damped and undamped
When linearizations can be trusted and when they shouldn't be trusted & nbsp;
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 The Heat and Wave Equations

Rigging  on  to get a pure sine fast Fourier fit of  on  
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.