Differential Equations for Engineers 

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Course Overview

Introduction to Differential Equations

Lecture 1

Week One Introduction

Euler Method

Lecture 2

Separable First-order Equations

Lecture 3

Separable First-order Equation: Example

Lecture 4

Linear First-order Equations

Lecture 5

Linear First-order Equation: Example

Lecture 6

Application: Compound Interest

Lecture 7

Application: Terminal Velocity

Lecture 8

Application: RC Circuit

Lecture 9

The SIR Model

The Basic Reproductive Ratio

Solution of the SIR Model

Welcome and Course Information

How to Write Math in the Discussions Using MathJax

Runge-Kutta Methods

Separable First-order Equations

Separable First-order Equation Examples

Linear First-order Equations

Change of Variables Transforms a Nonlinear to a Linear Equation

Linear First-order Equation: Examples

Saving for Retirement

Borrowing for a Mortgage

Terminal Velocity of a Skydiver

The Current in an RC Circuit

Diagnostic Quiz

Classify Differential Equations

Separable First-order ODEs

Linear First-order ODEs

Applications

Week One Assessment

Week Two Introduction

Euler Method for Higher-order ODEs

Lecture 10

The Principle of Superposition

Lecture 11

The Wronskian

Lecture 12

Homogeneous Second-order ODE with Constant Coefficients

Lecture 13

Case 1: Distinct Real Roots

Lecture 14

Case 2: Complex-Conjugate Roots (Part A)

Lecture 15

Case 2: Complex-Conjugate Roots (Part B)

Lecture 16

Case 3: Repeated Roots (Part A)

Lecture 17

Case 3: Repeated Roots (Part B)

Lecture 18

Complex Numbers

Second-order Equation as System of First-order Equations

Second-order Runge-Kutta Method

Linear Superposition for Inhomogeneous ODEs

Wronskian of Exponential Function

Roots of the Characteristic Equation

Distinct Real Roots

Hyperbolic Sine and Cosine Functions

Do You Know Complex Numbers?

Complex-Conjugate Roots

Sine and Cosine Functions

Repeated Roots

Superposition, the Wronskian, and the Characteristic Equation

Homogeneous Equations

Week Two Assessment

Week Three Introduction

Inhomogeneous Second-order ODE

Lecture 19

Inhomogeneous Term: Exponential Function

Lecture 20

Inhomogeneous Term: Sine or Cosine (Part A)

Lecture 21

Inhomogeneous Term: Sine or Cosine (Part B)

Lecture 22

Inhomogeneous Term: Polynomials

Lecture 23

Resonance

Lecture 24

RLC Circuit

Lecture 25

Mass on a Spring

Lecture 26

Pendulum

Lecture 27

Damped Resonance

Lecture 28

Nondimensionalization

Multiple Inhomogeneous Terms

Exponential Inhomogeneous Term

Sine or Cosine Inhomogeneous Term

Polynomial Inhomogeneous Term

When the Inhomogeneous Term is a Solution of the Homogeneous Equation

Do You Know Dimensional Analysis?

Another Nondimensionalization of the RLC Circuit Equation

Another Nondimensionalization of the Mass on a Spring Equation

Find the Amplitude of Oscillation

Solving Inhomogeneous Equations

Particular Solutions

Applications and Resonance

Week Three Assessment

Week Four Introduction

Definition of the Laplace Transform

Lecture 29

Laplace Transform of a Constant Coefficient ODE

Lecture 30

Solution of an Initial Value Problem

Lecture 31

The Heaviside Step Function

Lecture 32

The Dirac Delta Function

Lecture 33

Solution of a Discontinuous Inhomogeneous Term

Lecture 34

Solution of an Impulsive Inhomogeneous Term

Lecture 35

The Series Solution Method

Lecture 36

Series Solution of the Airy's Equation (Part A)

Lecture 37

Series Solution of the Airy's Equation (Part B)

Lecture 38

The Laplace Transform of Sine

Laplace Transform of an ODE

Solution of an Initial Value Problem

Heaviside Step Function

The Dirac Delta Function

Discontinuous Inhomogeneous Term

Impulsive Inhomogeneous Term

Series Solution Method

Series Solution of a Nonconstant Coefficient ODE

Solution of the Airy's Equation

The Laplace Transform Method

Discontinuous and Impulsive Inhomogeneous Terms

Series Solutions

Week Four Assessment

Week Five Introduction

Systems of Homogeneous Linear First-order ODEs

Lecture 39

Distinct Real Eigenvalues

Lecture 40

Complex-Conjugate Eigenvalues

Lecture 41

Phase Portraits

Lecture 42

Stable and Unstable Nodes

Lecture 43

Saddle points

Lecture 44

Spirals

Lecture 45

Coupled Oscillators

Lecture 46

Normal Modes (Eigenvalues)

Lecture 47

Normal Modes (Eigenvectors)

Lecture 48

Matrices and Determinants

Eigenvalues and Eigenvectors

Do You Know Matrix Algebra?

Eigenvalues of a Symmetric Matrix

Distinct Real Eigenvalues

Complex-Conjugate Eigenvalues

Phase Portraits

Nodes

Saddle Points

Spirals

Coupled Oscillators

Normal Modes of Coupled Oscillators

Systems of Differential Equations

Phase portraits

Normal Modes

Week Five Assessment

Week Six Introduction

Fourier Series

Lecture 49

Fourier Sine and Cosine Series

Lecture 50

Fourier Series: Example

Lecture 51

The Diffusion Equation

Lecture 52

Solution of the Diffusion Equation: Separation of Variables

Lecture 53

Solution of the Diffusion Equation: Eigenvalues

Lecture 54

Solution of the Diffusion Equation: Fourier Series

Lecture 55

Diffusion Equation: Example

Lecture 56

Partial Derivatives

Concluding Remarks

Fourier Series

Fourier series at x=0

Fourier Series of a Square Wave

Do You Know Partial Derivatives?

Nondimensionalization of the Diffusion Equation

Boundary Conditions with Closed Pipe Ends

ODE Eigenvalue Problems

Solution of the Diffusion Equation with Closed Pipe Ends

Concentration of a Dye in a Pipe with Closed Ends

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Acknowledgements

Fourier Series

Separable Partial Differential Equations

The Diffusion Equation

Week Six Assessment