ODE

Nov 14, 2023 ODE Foundations 26 min read

Ordinary Differential Equations (9): Chaos Theory and the Lorenz System

Deterministic yet unpredictable: the Lorenz system, butterfly effect, Lyapunov exponents, strange attractors, and the routes from order to chaos -- with Python simulations throughout.

Oct 28, 2023 ODE Foundations 22 min read

Ordinary Differential Equations (8): Nonlinear Systems and Phase Portraits

Step beyond linearity: predator-prey oscillations, competition exclusion, Van der Pol limit cycles, Hamiltonian systems, and the Poincaré-Bendixson theorem -- the full toolkit for nonlinear 2D dynamics.

Oct 11, 2023 ODE Foundations 24 min read

Ordinary Differential Equations (7): Stability Theory

Will a bridge survive the wind? Will an ecosystem recover from a shock? Stability theory answers these questions using Lyapunov functions, linearization, and bifurcation analysis.

Sep 24, 2023 ODE Foundations 22 min read

Ordinary Differential Equations (6): Linear Systems and the Matrix Exponential

When multiple variables interact, you need systems of ODEs. Learn the matrix exponential, eigenvalue-based solutions, phase portrait classification, and applications to coupled oscillators and RLC circuits.

Sep 7, 2023 ODE Foundations 26 min read

Ordinary Differential Equations (5): Power Series and Special Functions

When elementary functions fail, power series step in. Learn the Frobenius method and meet the special functions of physics: Bessel, Legendre, Hermite, and Airy functions -- with Python visualizations.

Aug 21, 2023 ODE Foundations 26 min read

Ordinary Differential Equations (4): The Laplace Transform

The engineer's secret weapon: turn differential equations into algebra with the Laplace transform. Learn the key properties, partial fractions, transfer functions, and PID control basics.

Aug 4, 2023 ODE Foundations 28 min read

Ordinary Differential Equations (3): Higher-Order Linear Theory

From springs to RLC circuits, the full theory of higher-order linear ODEs: superposition, the Wronskian, characteristic equations, undetermined coefficients, variation of parameters, and the resonance phenomenon.

Jul 18, 2023 ODE Foundations 26 min read

Ordinary Differential Equations (2): First-Order Methods

Master the four main techniques for first-order ODEs: separation of variables, integrating factors, exact equations, and Bernoulli substitution -- with applications to finance, pharmacology, ecology, and circuits.

Jul 1, 2023 ODE Foundations 30 min read

Ordinary Differential Equations (1): Origins and Intuition

Why do differential equations exist? Starting from cooling coffee and swinging pendulums, build your first ODE intuition and solve one in Python.