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Differential equation (view source)
Revision as of 06:14, February 12, 2009
, 06:14, February 12, 2009no edit summary
In [[mathematics]], a '''differential equation''' is an [[equation]] that relates a [[function]] to one or more of its [[derivative]]s. 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.
In [[mathematics]], a '''differential equation''' is an [[equation]] that relates a [[function]] to one or more of its [[derivative]]s. 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.<ref>Edwards, Henry C. and Penney, David E.. ''Differential Equations and Boundary Value Problems'' 4th Edition. Upper Saddle River, NJ: Pearson, 2008</ref>
The former is simpler of the two, as it can be written in the normal form
<math>y^{(n)} = F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, ... , y^{(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 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.
<math>\frac{\partial y}{\partial u} = k\frac{\partial^2 y}{\partial v^2}</math>
Obviously, partial differential equations are much more complicated to solve.
==References==
<small><references/></small>
[[category:calculus]]
[[category:calculus]]