# Differential Equations

Courseware

Intended for engineering students and others who require a working knowledge of differential equations; included are techniques and applications of ordinary differential equations and an introduction to partial differential equations.

You may purchase a copy of our courseware without any instructional services such as mentor feedback or grading if you are simply seeking reference materials rather than enrolling for a class.

## Syllabus

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

##### DE.01 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 step 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 superposition principle.

##### DE.02 The Forced Oscillator Diffeq y”[t] + b y’[t] + c y[t] = f[t]

- The undamped unforced oscillator y ‘ ‘[t] + c y[t] = 0
- The damped unforced oscillator y ‘ ‘[t] + b y ‘[t] + c y[t] = 0
- The damped forced oscillator y ‘ ‘[t] + b y ‘[t] + c y[t] = f[t]
- Steady state and transients for forced damped oscillators
- Resonance and beating
- Euler Identity
- The characteristic equation
- Using convolution integrals to try to get formulas for solutions of the forced oscillator diffeq
- Forcing an oscillator with a Dirac Delta function; the physical meaning of the impulse hit
- Amplitude and frequency of unforced oscillators
- Underdamped, critically damped, and overdamped oscillators
- Boundary value problems

##### DE.03 Laplace Transform and Fourier Analysis

- The Laplace transforms of a function y[t]
- How to write down the Laplace transform of the solution of a forced oscillator diffeq
- Solving forced oscillator diffeqs by inverting Laplace transforms
- Fast Fourier point fit and Fourier integral fit
- Combining Fourier fir and the Laplace transform to come up with good approximate formulas for period
- Fourier analysis for detecting resonance

#### Part 2: Introduction to Modern Theory of Differential Equations

##### 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 through flow plots
- Solving diffeq’s numerically with Mathematica
- Systems of interacting differential equations: The predator-prey model
- Sensitive dependence 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 there 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 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: 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 analysis of the eigenvalues of the Jacobian
- The pendulum oscillator: damped and undamped
- When linearization can be trusted and when it shouldn’t be trusted
- Linearization of pendulum oscillators: Using linearization 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 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.