forcing function (differential equations)

{{Short description|Function that only depends on time}}

{{About|a mathematical concept||Forcing function (disambiguation){{!}}Forcing function}}

In a system of differential equations used to describe a time-dependent process, a forcing function is a function that appears in the equations and is only a function of time, and not of any of the other variables.{{Cite web| url=http://depts.washington.edu/rfpk/training/tutorials/modeling/part8/10.html |title=How do Forcing Functions Work? |publisher=University of Washington Departments |url-status=dead |archive-url=https://web.archive.org/web/20030920030857/http://depts.washington.edu/rfpk/training/tutorials/modeling/part8/10.html |archive-date=September 20, 2003 }}{{Cite web |author=Packard A. |url=http://jagger.berkeley.edu/~pack/me132/Section7.pdf |url-status=dead | archive-url=https://web.archive.org/web/20170921193859/http://jagger.berkeley.edu/~pack/me132/Section7.pdf |archive-date=September 21, 2017 |format=PDF |title=ME 132 |date=Spring 2005 |publisher=University of California, Berkeley |page=55}} In effect, it is a constant for each value of t.

In the more general case, any nonhomogeneous source function in any variable can be described as a forcing function, and the resulting solution can often be determined using a superposition of linear combinations of the homogeneous solutions and the forcing term.{{cite book |last=Haberman |first=Richard |url=https://archive.org/details/elementaryapplie0000habe |title=Elementary Applied Partial Differential Equations |publisher=Prentice-Hall |year=1983 |isbn=0-13-252833-9 |page=272 |url-access=registration}}

For example, $f(t)$ is the forcing function in the nonhomogeneous, second-order, ordinary differential equation:

$$ay\text{'}\text{'}\; +\; by\text{'}\; +\; cy\; =\; f(t)$$