Ordinary Differential Equations
Apr 15, 2024
ODE Foundations
24 min readOrdinary Differential Equations (18): Frontiers and Series Finale
The series finale. Survey four research frontiers reshaping how we model dynamics -- Neural ODEs, delay equations, stochastic differential equations, and fractional calculus -- then take stock of the entire 18-chapter …
Mar 29, 2024
ODE Foundations
22 min readOrdinary Differential Equations (17): Physics and Engineering Applications
See ODEs in action across physics and engineering. Walk through the nonlinear pendulum, RLC circuit and resonance, Kepler orbits and conservation laws, multi-DOF structural vibration with tuned mass dampers, and fluid …
Mar 12, 2024
ODE Foundations
22 min readOrdinary Differential Equations (16): Fundamentals of Control Theory
Learn how differential equations power control systems. Cover transfer functions, PID controllers, root locus, Bode plots, state-space methods, controllability, observability, pole placement, LQR optimal control, and …
Feb 24, 2024
ODE Foundations
26 min readOrdinary Differential Equations (15): Population Dynamics
Mathematical ecology from single-species to spatial: Malthus, logistic, Allee, Lotka-Volterra predator-prey and competition, age-structured Leslie matrices, metapopulations, and Fisher-KPP traveling waves.
Feb 7, 2024
ODE Foundations
22 min readOrdinary Differential Equations (14): Epidemic Models and Epidemiology
Mathematical epidemiology from first principles. Build the SIR and SEIR models, derive R0 and the herd-immunity threshold, fit COVID-style scenarios with asymptomatic transmission and time-varying interventions.
Jan 21, 2024
ODE Foundations
24 min readOrdinary Differential Equations (13): Introduction to Partial Differential Equations
Step from ODEs into partial differential equations. Classify PDEs into parabolic, hyperbolic, and elliptic types. Solve the heat, wave, and Laplace equations using separation of variables and finite differences.
Jan 4, 2024
ODE Foundations
26 min readOrdinary Differential Equations (12): Boundary Value Problems
Boundary value problems specify the solution at both ends of an interval. Master shooting, finite differences, collocation, and Sturm-Liouville eigenproblems -- with applications from beam deflection to the quantum …
Dec 18, 2023
ODE Foundations
26 min readOrdinary Differential Equations (11): Numerical Methods
From Euler's tangent step to Dormand-Prince adaptive integrators: a working numerics toolkit. Convergence orders, A-stability, stiffness, and when to reach for Radau or BDF instead of RK45.
Dec 1, 2023
ODE Foundations
32 min readOrdinary Differential Equations (10): Bifurcation Theory
Bifurcation theory explains how smooth parameter changes cause dramatic qualitative shifts in system behavior. Master saddle-node, transcritical, pitchfork, and Hopf bifurcations through normal forms, stability …
Nov 14, 2023
ODE Foundations
26 min readOrdinary 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 readOrdinary 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 readOrdinary 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 readOrdinary 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 readOrdinary 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 readOrdinary 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 readOrdinary 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 readOrdinary 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 readOrdinary 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.