Differential Equations


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.


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.