ODE Foundations
From classical ODE methods to neural ODEs.
01Ordinary Differential Equations (1): Origins and Intuition
Why do differential equations exist? Starting from cooling coffee and swinging pendulums, build your first ODE intuition …
02Ordinary Differential Equations (2): First-Order Methods
Master the four main techniques for first-order ODEs: separation of variables, integrating factors, exact equations, and …
03Ordinary Differential Equations (3): Higher-Order Linear Theory
From springs to RLC circuits, the full theory of higher-order linear ODEs: superposition, the Wronskian, characteristic …
04Ordinary Differential Equations (4): The Laplace Transform
The engineer's secret weapon: turn differential equations into algebra with the Laplace transform. Learn the key …
05Ordinary 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 …
06Ordinary 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, …
07Ordinary Differential Equations (7): Stability Theory
Will a bridge survive the wind? Will an ecosystem recover from a shock? Stability theory answers these questions using …
08Ordinary Differential Equations (8): Nonlinear Systems and Phase Portraits
Step beyond linearity: predator-prey oscillations, competition exclusion, Van der Pol limit cycles, Hamiltonian systems, …
09Ordinary Differential Equations (9): Chaos Theory and the Lorenz System
Deterministic yet unpredictable: the Lorenz system, butterfly effect, Lyapunov exponents, strange attractors, and the …
10Ordinary Differential Equations (10): Bifurcation Theory
Bifurcation theory explains how smooth parameter changes cause dramatic qualitative shifts in system behavior. Master …
11Ordinary Differential Equations (11): Numerical Methods
From Euler's tangent step to Dormand-Prince adaptive integrators: a working numerics toolkit. Convergence orders, …
12Ordinary Differential Equations (12): Boundary Value Problems
Boundary value problems specify the solution at both ends of an interval. Master shooting, finite differences, …
13Ordinary Differential Equations (13): Introduction to Partial Differential Equations
Step from ODEs into partial differential equations. Classify PDEs into parabolic, hyperbolic, and elliptic types. Solve …
14Ordinary Differential Equations (14): Epidemic Models and Epidemiology
Mathematical epidemiology from first principles. Build the SIR and SEIR models, derive R0 and the herd-immunity …
15Ordinary Differential Equations (15): Population Dynamics
Mathematical ecology from single-species to spatial: Malthus, logistic, Allee, Lotka-Volterra predator-prey and …
16Ordinary Differential Equations (16): Fundamentals of Control Theory
Learn how differential equations power control systems. Cover transfer functions, PID controllers, root locus, Bode …
17Ordinary Differential Equations (17): Physics and Engineering Applications
See ODEs in action across physics and engineering. Walk through the nonlinear pendulum, RLC circuit and resonance, …
18Ordinary Differential Equations (18): Frontiers and Series Finale
The series finale. Survey four research frontiers reshaping how we model dynamics -- Neural ODEs, delay equations, …