Differential Equations

Master the mathematics of change and dynamics through interactive visualizations of slope fields, phase portraits, and dynamical systems

Part 1

Foundations

The language and intuition of differential equations

What is a Differential Equation?

Equations involving rates of change and their solutions

Slope Fields

Visualizing solutions without solving

Separation of Variables

The first solving technique

Exponential Growth and Decay

The simplest and most important differential equation

Part 2

First-Order Techniques

Mastering the toolkit for first-order equations

Linear First-Order Equations

Integrating factors and the systematic approach

Exact Equations

When a differential equation is secretly a total derivative

Substitution Methods

Transforming hard problems into solvable ones

Existence and Uniqueness

When do solutions exist and when are they unique

Part 3

Second-Order Linear Equations

The backbone of physics and engineering

Second-Order Homogeneous Equations

Characteristic equations and the nature of solutions

Complex Roots and Oscillations

When solutions oscillate and decay

Nonhomogeneous Equations

Undetermined coefficients and variation of parameters

Mechanical Vibrations

Springs, damping, and resonance

Part 4

Systems of Differential Equations

Multiple interacting quantities and their dynamics

Introduction to Systems

From single equations to coupled systems

Phase Portraits

The geometry of dynamical systems

The Eigenvalue Method

Solving linear systems using linear algebra

Stability and Classification

Nodes, spirals, saddles, and centers

Part 5

Advanced Methods

Powerful techniques for harder problems

The Laplace Transform

Transforming differential equations into algebra

Inverse Laplace and Applications

Solving initial value problems with transforms

Series Solutions

Power series methods for differential equations

Numerical Methods

Euler, Runge-Kutta, and computational approaches

Part 6

Partial Differential Equations

Functions of multiple variables and their applications

Introduction to PDEs

Heat, wave, and Laplace equations

The Heat Equation

Diffusion, smoothing, and separation of variables

The Wave Equation

Vibrating strings and wave propagation

Fourier Series

Decomposing functions into frequencies