Differential equation
From Mathematics
A vast subject within the mathematical field. It can most simply be defined, for a layman, as any equation that involves any combination of the following:
- An independent variable (x)
- Functions of the independent variable (or Dependent Variables) (g(x))
- A primary dependent variable ("THE function in question") (y(x))
- And, necessarily, any number of and degrees of derivatives of the primary function (y′(x), y″(x))
An example Differential Equation (DE) is as follows:
The concept is simple: We are relating the trends in the function to other trends in the function. That is to say, a DE relates derivatives of varying degrees (slopes, gradients) to one another and to other functions of x.
The study of differential equations is an extension onto differential and integral calculus.
[edit] Objective
Within the world of academia, when given a DE problem, the objective is often to find the function y(x) as an explicit function in terms of only x, and possible functions of x; sometimes only implicit functions of y(x) are possible; but necessarily the objective is to remove all derivatives of y(x) from the equation.
For example, the DE:
Will have the following solution:
[edit] Application
DE's are highly applicable to the real world. Often times, be in in statistics, economics or theoretical physics, data will correlate in such ways that determining a DE is more easily done than trying to extrapolate a direct function. Most often, some of the oddest looking explicit functions had to be arrived at through DE analysis.
