# 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 