Instruct: Given a question, retrieve relevant passages that answer the question.
Query: Who created differential equations?
Differential equations first came into existence with the invention of calculus by Newton and Leibniz. In Chapter 2 of his 1671 work "Methodus fluxionum et Serierum Infinitarum",[1] Isaac Newton listed three kinds of differential equations:
Ordinary differential equations occur in many scientific disciplines, for instance in physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation
where "f(t)" is a periodic function. It is named after George William Hill, who introduced it in 1886.
According to Mary Croarken in her paper "Computing in Britain During World War II," by 1945, the Cambridge Mathematical Laboratory created by John Lennard-Jones utilized the latest computing devices to perform the equations. These devices included a model "differential analyser," and the Mallock machine, described as "an electrical simultaneous equation solver." According to Croarken, the Ministry was also interested in the new arrival of a differential analyzer accommodating eight integrators. This exotic computing device built by Metropolitan-Vickers in 1939 consisted of wheel and disk mechanisms that could provide descriptions and solutions for differential equations. Output resulted in a plotted graph.
Starting in 1927, Bush constructed a differential analyzer, an analog computer that could solve differential equations with as many as 18 independent variables. This invention arose from previous work performed by Herbert R. Stewart, one of Bush's masters students, who at Bush's suggestion created the integraph, a device for solving first-order differential equations, in 1925. Another student, Harold Hazen, proposed extending the device to handle second-order differential equations. Bush immediately realized the potential of such an invention, for these were much more difficult to solve, but also quite common in physics. Under Bush's supervision, Hazen was able to construct the differential analyzer, a table-like array of shafts and pens that mechanically simulated and plotted the desired equation. Unlike earlier designs that were purely mechanical, the differential analyzer had both electrical and mechanical components. Among the engineers who made use of the differential analyzer was General Electric's Edith Clarke, who used it to solve problems relating to electric power transmission. For developing the differential analyzer, Bush was awarded the Franklin Institute's Louis E. Levy Medal in 1928.
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example Riccati equation).
Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on.[11][12] Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.
Unlike ODEs, which are equations containing a function of one variable and its derivative for the same input, functional differential equations are equations containing a function and some of its derivatives for different input values.The simplest type of FDE, called the retarded functional differential equation (RDE) or retarded differential difference equation, is of the form
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
This is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. In this article, only ordinary differential equations are considered.
Complex differential equation Exact differential equation Functional differential equation Initial condition Integral equations Numerical methods for solving differential equations Picard–Lindelöf theorem on existence and uniqueness of solutions Recurrence relation, also known as 'difference equation' Abstract differential equation
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 the 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, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.
The researchers developed a general continuum theory for a bundle of hairs, treating each hair as an elastic filament with random intrinsic curvature. From this they created a differential equation for the shape of the bundle relating the elasticity, gravity, and orientational disorder and extracted a simple equation of state to relate the swelling pressure to the measured random curvatures of individual hairs. The equation itself is a fourth order non linear differential equation.
where:The relationship between a PDAE and a partial differential equation (PDE) is analogous to the relationship between an ordinary differential equation (ODE) and a differential algebraic equation (DAE).
A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (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.
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.
A differential equation is a relation between a collection of functions and their derivatives. An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A partial differential equation is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, Newton's second law, which describes the relationship between acceleration and force, can be stated as the ordinary differential equation
The heat equation in one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation
Here is the temperature of the rod at position and time and is a constant that depends on how fast heat diffuses through the rod.
See also Nonlinear partial differential equation, List of partial differential equation topics and List of nonlinear ordinary differential equations.
A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (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 by hand, or used to create a relevant computer model.
A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times. A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations. A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.
is called a Bernoulli differential equation where formula_2 is any real number and formula_3 and formula_4. It is named after Jacob Bernoulli who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli equation is the logistic differential equation.
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 the approximation of the solution of a differential equation by the solution of a corresponding difference equation.
All second order differential equations with constant coefficients can be transformed into their respective canonic forms. This equation is one of these three cases: Elliptic partial differential equation, Parabolic partial differential equation and Hyperbolic partial differential equation.
FDE is the general form for more specific types of differential equations that are used in numerous applications. There are delay differential equations, integro-differential equations, and so on.
A "solution" of a differential equation is a function that satisfies the equation.
The solutions of a homogeneous linear differential equation form a vector space. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.