What Are Differential Equations?

Date: 12/17/2003 at 15:49:57
From: Adam
Subject: What are differential equations?

Hi.  I would like to know what exactly differential equations are.  I 
understand they are used for modeling, but how do they derive the 
formulas?  Why does it have two variables: (x,y)?  Why do they always 
have dy/dx in front of every formula?  And finally, how come the 
solutions are formulas and not numbers? 

Date: 12/23/2003 at 10:39:38
From: Doctor George
Subject: Re: What are differential equations?

Hi Adam,

Differential equations are equations in which we know some
relationship between the derivatives of a function.  The relationship 
can include the function itself.  The goal is to determine what
function satisfies this relationship. 

The use of dy/dx denotes the derivative of y(x) with respect to x. 
This notation was introduced by Leibniz.  Another notation for the 
derivative is f'(x) where y = f(x).  The second derivative (the 
derivative of the derivative) is denoted by f''(x), or d^2y/dx^2.  In 
general, the nth derivative is denoted d^ny/dx^n.

You may already know one of the most common differential equations:

           d^2x
           ---- = g
           dt^2

where x is the height of an object and g is its acceleration due to 
the force of gravity.  This is also one of the simplest differential 
equations to solve, as it is a matter of straightforward integration.

           dx
           -- = g*t + v
           dt

Here v is a constant of integration representing the initial velocity
of the object.  If we integrate again we get

           x(t) = (g/2)*t^2 + v*t + h

where h is another constant of integration representing the initial 
height of the object.  If you take the second derivative of x(t) with 
respect to t you will find that this equation satisfies or solves the
differential equation.  In other words, taking the second derivative
of x(t) with respect to t will return you to

           d^2x
           ---- = g
           dt^2

Another common and fairly simple differential equation is

           dx
           -- = -Abx(t)
           dt

You can probably see that the following is a solution.

           x(t) = Ae^(-bt)

This is an equation for exponential decay.

Note that since the differential equation involves derivatives, you 
will be using integration as a solving technique, trying to work 
backwards and discover what the original function was that leads to
the given derivative.  Because we are finding a function, answers tend
to be functions and not numbers, as you noted.

Differential equations show up most anywhere that rates of change 
are being measured, as rates are often expressed as changes with
respect to time.  The subject can become very complex at an advanced
level, but I hope that gives you a good start.  Write again if you
need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/