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Differential equation
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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.
A simplified real world example of a differential equation is modeling the acceleration of a ball falling through the air (considering only gravity and air resistance).
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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.
A simplified real world example of a differential equation is modeling the acceleration of a ball falling through the air (considering only gravity and air resistance). The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is a constant but air resistance is proportional to the ball's velocity. This means the ball's acceleration is dependent on its velocity. Because acceleration is the derivative of velocity, solving this problem requires a differential equation.
Differential equations arise in many areas of science and technology; whenever a deterministic relationship involving some continuously changing quantities (modeled by functions) and their rates of change (expressed as derivatives) is known or postulated. This is well illustrated by classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's Laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In many cases, this differential equation may be solved explicitly, yielding the law of motion.
Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. Only the simplest differential equations admit solutions given by explicit formulas. Many properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
Directions of study
The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. 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. Instead, solutions can be approximated using numerical methods.
Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolic type.
The study of the stability of solutions of differential equations is known as stability theory.
Types of differential equations
- An ordinary differential equation (ODE) is a differential equation in which the unknown function is a function of a single independent variable.
- A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and their partial derivatives.
- A delay differential equation (DDE) is a differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
- A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.
- A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.
Each of those categories is divided into linear and nonlinear subcategories. A differential equation is linear if the dependent variable and all its derivatives appear to the power 1 and there are no products or functions of the dependent variable. Otherwise the differential equation is nonlinear. Thus if u′ denotes the first derivative of the function u, then the equation
is linear, while the equation
is nonlinear. Solutions for a linear equation where the unknown function or its derivative(s) appear in each term (linear homogeneous equations), can be added together or multiplied by an arbitrary constant to obtain additional solutions to that equation. However, there is no general way to obtain families of solutions of nonlinear equations, except when they exhibit symmetries. (See symmetries and invariants). Linear equations frequently appear as approximations to nonlinear equations, and these approximations are only valid under restricted conditions.
Another important characteristic of a differential equation is its order, which is the order of the highest derivative (of a dependent variable) appearing in the equation. For instance, a first-order differential equation contains only first derivatives, like both examples above.
Connection to difference equations
The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.
See also: Time scale calculus.
Differential equations are also divided into homogeneous and heterogeneous equations.
Homogeneous equations are equations where the variables are not separated easily. In heterogeneous equations the variables can be separated with no difficulty in order to solve the equations.
Universality of mathematical description
Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. 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 propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, whose theory was developed by Joseph Fourier, is governed by another second order partial differential equation, the heat equation. It turned out that many diffusion processes, while seemingly different, are described by the same equation; Black-Scholes equation in finance is for instance, related to the heat equation.
Notable differential equations
Biology
- Verhulst equation - biological population growth
- Lotka-Volterra equations - biological population dynamics
- Replicator dynamics - may be found in theoretical biology
Economics
See also
External links
- MIT Open CourseWare video
- Paul Dawkins, Lamar University
- , S.O.S. Mathematics
- Introduction to modeling by means of differential equations, with critical remarks.
- Java applet tool used to solve differential equations.
- online solving first order (linear and with separated variables) and second order linear differential equations (with constant coefficients), including intermediate steps in the solution, powered by Maxima
- , from the Mathematical Association of America
- MATLAB models
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- Brings together all the material on differential equations from Plus, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge.
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