ODE
Ordinary 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 …
Ordinary 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 …
Ordinary 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 …
Ordinary 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.
Ordinary 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.
Ordinary 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.
Ordinary 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 …
Ordinary 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.
Ordinary 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 …
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.