14-11-2012, 12:37 PM
DECAY AND GROWTH OF DIFFERENTIAL EQATION AND USES OF TAYLOR THEORM
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Taylor's theorem
The exponential function y = ex (continuous red line) and the corresponding Taylor polynomial of degree four around the origin (dashed green line).
In calculus, Taylor's theorem gives a sequence of approximations of a differentiable function around a given point by polynomials (the Taylor polynomials of that function) whose coefficients depend only on the derivatives of the function at that point. The theorem also gives precise estimates on the size of the error in the approximation. The theorem is named after the mathematician Brook Taylor, who stated it in 1712, though the result was first discovered 41 years earlier in 1671 by James Gregory.
Estimates of the remainder
Another common version of Taylor's theorem holds on an interval (a − r, a + r) where the variable x is assumed to take its values. This formulation of the theorem has the advantage that it is often possible to explicitly control the size of the remainder terms, and thus arrive at an approximation of a function valid in a whole interval with precise bounds on the quality of the approximation.
What is a differential equation ?
By studying the exponential functions, and picking a convenient base, we have inadvertantly stumbled on a relationship satisfied by the function and its derivative. That relationship is called a differential equation . The topic of differential equations is an extremely important one in mathematics and science, as well as many other branches of studies (economics, commerce) in which changes occur and in which predictions are desirable. In most such circumstances, the systems studied come with some kind of "Natural Laws", or observations that, when translated into the language of mathematics, become differential equations. It is then the job of the mathematician to try to figure out what are the predictions, i.e. to find the functions that satisfy those equations.
In our work we have been lucky enough to start with a function and show that there is a differential equation that it satisfies, .
DEFINITION
We say that a function is a solution to a differential equation if, when we plug it (and its various derivatives) into the equation, we find that the equation is satisfied.
Comment:
It is important to notice right off, that a solution to a differential equation is a function , unlike the solution to an algebraic equation which is (usually) a number, or a set of numbers. This makes differential equations much more interesting, and often more challenging to understand, than algebraic equations.