Everything about Differential Equation totally explained
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 of 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; 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 isn't 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, for example don't 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 don't 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
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 of a linear equation in which the unknown function or its derivative(s) appear in each term (
linear homogeneous equations) may be added together or multiplied by an arbitrary constant in order to obtain additional solutions of that equation, but there's 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.
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 brilliantly 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.
Famous differential equations
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