Ordinary Differential Equations
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 …