Direction Fields and Integral Curves

Euler's Method for Differential Equations

Sinusoidal Inputs

Exploration of the Euler's Method Applet

Euler's Numerical Method for y'=f(x,y) and its Generalizations

Direction Fields in Differential Equations

First-order Constant Coefficient Linear ODE's

Exploration of the Isoclines Applet

First-order Linear ODEs: Steady-state and Transient Solutions

Exploration of the Amplitude and Phase: First Order Applet

Sinusoidal Functions

First-Order Substitution Methods: Bernoulli and Homogeneous ODEs

Complex Numbers and Euler's Formula

First-order Autonomous ODEs: Qualitative Methods and Applications

Solutions of First-order Linear Equations

Complex Numbers and Complex Exponentials

Linear Equations in Differential Equations

First-order Linear Differential Equations with Constant Coefficients

Applications to Temperature, Mixing, RC-Circuit, Decay, and Growth Models

Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases

Complex Characteristic Roots; Undamped and Damped Oscillations

Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians

General Theory for Inhomogeneous ODE's; Stability Criteria for Constant-coefficient ODE's

Finding Particular Solutions to Inhomogeneous ODE's: Operator and Solution Formulas Involving Exponentials

Interpretation of the Exceptional Case: Resonance

Introduction to Fourier Series; Basic Formulas for Period 2π

More General Periods; Even and Odd Functions; Periodic Extension

Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds

Introduction to the Laplace Transform; Basic Formulas

Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's

Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems

Using Laplace Transform to Solve ODE's with Discontinuous Inputs