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An integro-differential equation (IDE) is an equation that combines aspects of a differential equation and an integral equation. 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 ...
An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory [ 5 ] [ 6 ] [ 7 ] Chrystal's equation
Examples of differential equations; Autonomous system (mathematics) Picard–Lindelöf theorem; Peano existence theorem; Carathéodory existence theorem; Numerical ordinary differential equations; Bendixson–Dulac theorem; Gradient conjecture; Recurrence plot; Limit cycle; Initial value problem; Clairaut's equation; Singular solution ...
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 ).
Differential equations play a prominent role in many scientific areas: mathematics, physics, engineering, chemistry, biology, medicine, economics, etc. This list presents differential equations that have received specific names, area by area.
For some differential equations, application of standard methods—such as the Euler method, explicit Runge–Kutta methods, or multistep methods (for example, Adams–Bashforth methods)—exhibit instability in the solutions, though other methods may produce stable solutions.
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 =.
Integro-differential equations model many situations from science and engineering, such as in circuit analysis. By Kirchhoff's second law, the net voltage drop across a closed loop equals the voltage impressed (). (It is essentially an application of energy conservation.)