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/
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