A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.
At any given time a dynamical system has a state given by a set of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). Small changes in the state of the system create small changes in the numbers. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic; in other words, for a given time interval only one future state follows from the current state.
The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. Once the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a trajectory or orbit.
Lec 27 | MIT 18.03 Differential Equations, Spring 2006
Lec 31 | MIT 18.03 Differential Equations, Spring 2006
System Dynamics and Control: Module 3 - Mathematical Modeling Part I
Multiagent Dynamical Systems
Fixed points and stability of a nonlinear system
Dynamical Systems
Lecture 6, Systems Represented by Differential Equations | MIT RES.6.007 Signals and Systems
State Space Representation ( Dynamic Systems ) | Mechanical Engineering
Logical Analysis of Hybrid Systems
Lecture 2 | Introduction to Linear Dynamical Systems
Lecture 3 | Introduction to Linear Dynamical Systems
Lecture 6 | Introduction to Linear Dynamical Systems
Lecture 9 | Introduction to Linear Dynamical Systems
Lecture 10 | Introduction to Linear Dynamical Systems
Lec 27 | MIT 18.03 Differential Equations, Spring 2006
Lec 31 | MIT 18.03 Differential Equations, Spring 2006
System Dynamics and Control: Module 3 - Mathematical Modeling Part I
Multiagent Dynamical Systems
Fixed points and stability of a nonlinear system
Dynamical Systems
Lecture 6, Systems Represented by Differential Equations | MIT RES.6.007 Signals and Systems
State Space Representation ( Dynamic Systems ) | Mechanical Engineering
Logical Analysis of Hybrid Systems
Lecture 2 | Introduction to Linear Dynamical Systems
Lecture 3 | Introduction to Linear Dynamical Systems
Lecture 6 | Introduction to Linear Dynamical Systems
Lecture 9 | Introduction to Linear Dynamical Systems
Lecture 10 | Introduction to Linear Dynamical Systems
Lecture 11 | Introduction to Linear Dynamical Systems
Lecture 12 | Introduction to Linear Dynamical Systems
Lecture 13 | Introduction to Linear Dynamical Systems
Lecture 15 | Introduction to Linear Dynamical Systems
Homogeneous Systems of Linear Equations - Intro to Eigenvalue/Eigenvector Method
Differential Equations - Solve Linear System using Laplace transforms
Linearizing non-linear dynamic equations
Partial differential equation
Differential Equations, The Exponential Map Perspective - Lecture 7
Lecture 1 | Introduction to Linear Dynamical Systems
Differential Equations: The Exponential Map Perspective - Lecture 1
Differential Equations: The Exponential Map Perspective - Lecture 6
Differential Equations, the Exponential Map Perspective - Lecture 11
Physics 111: Non-Linear Dynamics and Chaos (NLD)
Introduction to Nonlinear PDEs I. Nonlinear Diffusion Equation
All About - Dynamical systems theory
Differential equation
Interactive Exploration of a Dynamical System
Examining Dynamic Systems (Dr. Eric R. Kaufmann)
Dynamical Modeling with PottersWheel
Graphing Linear Equations2
Non Linear Eiffel Tower Warp
Lesson 6.5--Special Cases of Linear Systmes
Algebra 1: Fitting Equations to Data
Linear Applications 1
Linear Applications 2
Booklet-System Dynamisch (T-Systems)
Section 2.2-2.4-Solving Linear Equations
NOTES 05.7 Linearization Ex 6
Linear Measurement- Capacity