Skip to main content

ordinary differential equations

ordinary differential equations Differential equations that involve one independent variable, which in practice may be a space or time variable. Except in simple cases the solution cannot be determined analytically and approximation methods are used.

Numerical methods are mainly developed for equations involving first derivatives only, written in the form y′ = f(x,y), axb,

where y and f are s-component vectors with component functions yi(x), fi(x, y1(x), y2(x),…, ys(x))

Equations involving higher derivatives can be equivalently written in this form by introducing intermediate functions for the higher derivatives. Alternatively direct methods may be derived for such problems (see Nyström methods).

In general, s conditions must be imposed to determine a particular solution. If the values y(a) = y0 are specified, it is an initial-value problem. These problems can be solved directly using step-by-step methods, such as Runge-Kutta methods, linear multistep methods, or extrapolation methods, which determine approximations at a set of points in [a,b]. The problem is a boundary-value problem if the s conditions are given in terms of the component functions at a and b. In general, such problems require iterative methods, such as the shooting method. However, if f is linear in y, finite-difference methods can be advantageous. Excellent software has been developed for both types of problem.

An area of particular interest in many applications is the solution of stiff equations. A stiff system possesses solutions that decay very rapidly over an interval that is short relative to the range of integration, and the solution required varies slowly over most of the range. To allow large steps in the slowly varying phases, it is necessary to use special methods, such as the implicit trapezoidal rule: xn+1 = xn + h yn+1 = yn + ½h(f(xn+1, yn+1) + f(xn, yn))

At each step a system of equations has to be solved for yn+1, using very often a modification of Newton's method. More straightforward explicit methods rapidly lead to catastrophic error growth unless the stepsize h is prohibitively small. These problems are still the subject of very active research interest.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"ordinary differential equations." A Dictionary of Computing. . 15 Aug. 2018 <>.

"ordinary differential equations." A Dictionary of Computing. . (August 15, 2018).

"ordinary differential equations." A Dictionary of Computing. . Retrieved August 15, 2018 from

Learn more about citation styles

Citation styles gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, cannot guarantee each citation it generates. Therefore, it’s best to use citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

The Chicago Manual of Style

American Psychological Association

  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.