Term Papers on Differential Equations

Term Paper Title	Differential Equations
# of Words	2898
# of Pages (250 words per page double spaced)	11.59

Differential Equations

Introduction

     Calculus is one of the most powerful and useful branches in the field of mathematics. It has been a major contributor to our knowledge of physics and engineering. Even though this field is so vast and complicated, it is only sub-divided into two major groups. These two branches being Differential and Integral Calculus. Combining the branches, we are able to achieve a higher goal; finding the solution to a differential equation. First, the basis of differential and integral calculus must be explained. In order to understand the concepts involved in differential equations one must be familiarized with the derivative.
The Derivative
     The basic definition of the derivative is the rate of change of a function (Simmons 46). Rate of change can be used to figure out many different things. One of the most common uses for the derivatives is to find the sensitivity to changes. For example, using derivatives one can see how much change will occur if the variable changes slightly. To find the rate of change, one must first know the function. A function is a process being done or usually it is described by an equation. In an equation the function is usually stated as being f(x) or read as f at x (DuChateau 59). Using the function as your dependent variable you can simplify it by calling it y. The main mathematical equation used for find the derivative of a function is:

                                                Y =                D Y
                                         limDx®0  DX

     The term limit (lim.) is very important to Calculus. In the equation the limit was used to get the points between X1 and X2 as small as possible. So we want the DX (X2 - X1) to approach zero. This would give us a much more accurate reading. If DX was not limited to zero than we would get a lot of unwanted and vague information (DuChateau 62).
     The other way to describe the mathematical definition of the derivative is:
     Y=           F(x +Dx) – f(x)
                  lim.Dx®0          DX

f(x + Dx) is the final result of the y-function with the change and f(x) is the original or initial function. Subtracting f(x) from f(x + Dx) gives you DY (Simmons 46).
Integration
     After understanding this concept, one can now learn to integrate. Integration is the process where one knows the derivative and must find the original function. Basically, it is the accumulation of all the changes. To better understand this concept, consider a continuous curve y = f(x) lying above the x-axis and let (a , b) be an interval on the x-axis. We can then denote the function to be A(x). It has already been proven that A’(x) =  f(x). This proves that finding the area function reduces to doing the opposite of the differentiation process and recovering A(x) from its known derivative f(x). Once A(x) is found, the area under the curve
y = f(x) over a specific interval can be found by evaluating the specific y values (Anton 297).

            Y - axis
                                Y = f(x)

                                   A(x)

                    A                                 B          X - Axis


     Another proven fact about derivatives is that the derivative of a constant is zero. This causes a problem because one cannot find the exact function because it is not known if the original function ever contained a constant. If we suspect a constant was originally in the function we can then denote it writing + C at the end This usually occurs when the limit is between -¥ ® ¥. In other words, this interval contains every number. Functions that contain constants are called indefinite integrals. The definition explains the term very nicely because if we did not know whether the initial function had a constant or not, we would be “indefinite” in our conclusions (Anton 299).
     When dealing with integrals there are a few principals that must be used. For example, if

    d         [F(x)] = f(x)
   dx                        then the functions of the form F(x) + C are antiderivatives of f(x). This can be denoted by writing:  ò f(x) dx = F(x) + C
The symbol ò is called an integral sign. So the integral of a function is written ò f(x) and it...

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