Difference between revisions of "Differential equation"
| Line 17: | Line 17: | ||
<math>y = F(u,v)</math> | <math>y = F(u,v)</math> | ||
| − | then the derivatives of y in the ordinary equation may be [[partial derivative|partial derivatives]] with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. | + | then the derivatives of y in the ordinary equation may be [[partial derivative|partial derivatives]] with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. For example, the 1-dimensional [[wave equation]]<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref> : |
| − | <math>\frac{\partial y}{\partial | + | <math>\frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}</math> |
Obviously, partial differential equations are much more complicated to solve. | Obviously, partial differential equations are much more complicated to solve. | ||
Revision as of 05:25, February 27, 2009
In mathematics, a differential equation is an equation that relates a function to one or more of its derivatives. Differential equations are especially applicable when the tools of algebra, which are ideally suited for static systems, are not enough. Many physical systems are modeled by solving differential equations, although their usefulness extends well into other fields of science such as chemistry and economics.
Types of Differential Equations
There are two main types of differential equations: Ordinary Differential Equations and Partial Differential Equations.[1]
The former is simpler of the two, as it can be written in the normal form
<math>\frac{d^{(n)}y}{dx^{(n)}} = F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, ... , \frac{d^{(n-1)}y}{dx^{(n-1)}})</math>
for a simple function <math>y = g(x)</math>
The function F consists of the function y and its derivatives up to the nth order. Notice that y is comprised of only one independent variable x. A differential equation is considered ordinary if the function y in F is dependent on only one variable. It is important to note that, while most ordinary differential equations can be written in the normal form (isolating the highest derivative on one side of the equation and moving all other variables to the other), there are equations in which this cannot be done.
If y were a function of multiple variables, for example
<math>y = F(u,v)</math>
then the derivatives of y in the ordinary equation may be partial derivatives with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. For example, the 1-dimensional wave equation[2] :
<math>\frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}</math>
Obviously, partial differential equations are much more complicated to solve.