Differential Equation

A Differential Equation is a mathematical equation that relates an unknown differentiable function, one or more of its function variables and the function's derivatives.

• Context:
  • It can range from being an Ordinary Differential Equation to being a Partial Differential Equation.
  • It can be an input to a Differential Equation Solving System (solving a differential equation solving task).
• Example(s)
  • [math]\displaystyle{ u'' + p(x)u' + q(x)u=f(x),\quad x\gt a }[/math]
  • [math]\displaystyle{ \frac{\rd^3 y}{\rd x^3} - x \frac{\rd y}{\rd x} + (1-x)y = \sin y }[/math]
• Counter-Example(s):
  • Non-Differential Equation.
  • a Differentiable Function.
• See: First Derivative, Second Derivative.

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References

2012

• http://en.wikipedia.org/wiki/Differential_equation
  • 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.

    Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

• http://en.wikipedia.org/wiki/Differential_equation#Related_concepts
  • 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.

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