ODE
ODE Chapter 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.
ODE Chapter 8: Nonlinear Systems and Phase Portraits
Step beyond linearity: predator-prey oscillations, competition exclusion, Van der Pol limit cycles, Hamiltonian systems, and the Poincare-Bendixson theorem -- the full toolkit for nonlinear 2D dynamics.
ODE Chapter 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.
ODE Chapter 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.
ODE Chapter 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.
ODE Chapter 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.
ODE Chapter 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.
ODE Chapter 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.
ODE Chapter 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.