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In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
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.
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations.
Name Order Equation Applications Airy: 2 = [1] Optics: Bessel: 2 + + = Wave propagation: Cauchy-Euler: n () + () + + =Chebyshev: 2 ...
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
Functional differential equations of neutral type, or neutral differential equations occur when {,, …,} =. Neutral differential equations depend on past and present values of the function, similarly to retarded differential equations, except it also depends on derivatives with delays.
The study of these differential equations with constant coefficients dates back to Leonhard Euler, who introduced the exponential function e x, which is the unique solution of the equation f′ = f such that f(0) = 1. It follows that the n th derivative of e cx is c n e cx, and this allows solving homogeneous linear differential equations ...
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.)