Ordinary Differential Equations

Ordinary Differential Equations Describe Evolving Systems

How an ODE turns a rule for change plus initial information into a trajectory through time.

First-Order ODEs Model Growth, Decay, and Balance

A foundational guide to slope fields, separable equations, linear equations, and equilibrium solutions.

Requires: Ordinary Differential Equations Describe Evolving Systems

Second-Order ODEs Govern Oscillation and Motion

How characteristic roots, initial conditions, damping, and forcing shape second-order systems.

Requires: Ordinary Differential Equations Describe Evolving Systems

Systems of ODEs Track Interacting Quantities

How coupled first-order equations, phase planes, equilibria, and eigenvalues describe multivariable dynamics.

Requires: First-Order ODEs Model Growth, Decay, and Balance

Slope Fields Show an ODE Before It Is Solved

How short line segments visualize the local direction rule in a first-order differential equation.

Requires: First-Order ODEs Model Growth, Decay, and Balance

Characteristic Roots Classify Linear ODE Solutions

How an algebraic polynomial determines growth, decay, and oscillation in constant-coefficient equations.

Requires: Second-Order ODEs Govern Oscillation and Motion

Phase Planes Show the Geometry of Two-State Systems

How trajectories and vector fields reveal coupled dynamics without plotting time directly.

Requires: Systems of ODEs Track Interacting Quantities

Euler’s Method Follows an ODE One Step at a Time

How local slopes generate a numerical approximation and why step size controls error.

Requires: Slope Fields Show an ODE Before It Is Solved

Separable ODEs Put Each Variable on Its Own Side

How separation and integration solve first-order equations with factorable rates.

Requires: First-Order ODEs Model Growth, Decay, and Balance

Integrating Factors Solve Linear First-Order ODEs

How one multiplier turns a linear differential equation into a product derivative.

Requires: First-Order ODEs Model Growth, Decay, and Balance

Equilibria Organize Autonomous ODEs

How constant solutions and phase lines reveal stability without solving an equation explicitly.

Requires: First-Order ODEs Model Growth, Decay, and Balance

Damping Determines How Oscillations Fade

How underdamped, critically damped, and overdamped systems return toward equilibrium.

Requires: Characteristic Roots Classify Linear ODE Solutions

Periodic Forcing Can Produce Resonance

How external forcing creates a steady response and why near-natural frequencies can amplify motion.

Requires: Second-Order ODEs Govern Oscillation and Motion

Eigenvalues Classify Linear System Stability

How the real and imaginary parts of eigenvalues predict growth, decay, and rotation near equilibrium.

Requires: Systems of ODEs Track Interacting Quantities