Such relations are common in mathematical models and scientific laws; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
The theory of dynamical systems analyzes the qualitative aspects of solutions, such as their average behavior over a long time interval.
Differential equations came into existence with the invention of calculus by Isaac Newton and Gottfried Leibniz.
[3] This is an ordinary differential equation of the form for which the following year Leibniz obtained solutions by simplifying it.
[4] Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.
[9] The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem.
In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat),[10] in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures.
This partial differential equation is now a common part of mathematical physics curriculum.
In classical mechanics, the motion of a body is described by its position and velocity as the time value varies.
Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.
Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity.
Finding the velocity as a function of time involves solving a differential equation and verifying its validity.
Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous.
This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.
(This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.)
PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.
PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics.
These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs.
Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of chaos.
Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf.
However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.
Suppose we had a linear initial value problem of the nth order: such that For any nonzero
The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering.
Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons.
Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions.
In biology and economics, differential equations are used to model the behavior of complex systems.
However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations.
Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena.
As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond.
The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic.