| 单词 | differential equation |
| 释义 | differential equation Ordinary and partial differential equations are hugely important in mathematical models describing phenomena in the real world (see Bessel's equation, Black-Scholes equation, calculus of variations, Euler's equation, heat equation, Laplace's equation, Maxwell's equations, Navier-Stokes equations, predator-prey equations, Schrödinger's equation, wave equation). The problem of solving a differential equation is to find the general form of functions y whose derivatives satisfy the equation. In certain circumstances, it can be shown that a differential equation of order n has a general solution for y, involving n arbitrary constants. Such examples include linear differential equations with constant coefficients. But this is far from the general case (see Picard's theorem). Further information, such as initial conditions or boundary conditions, may be given to specify uniquely a solution within the general solution. For example, y″– ω2y = 0, models simple harmonic motion, and has general solution y(x) = Acosωx + Bsinωx. If further we know y(0) = 2 and y′(0) = 1, then this specifies the solution uniquely as y(x) = 2cosωx + ω–1sinωx. See homogeneous first-order differential equation, linear differential equations, linear first-order differential equation, Laplace transform, separable first-order differential equation. |
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