Variation of parameters: Difference between revisions

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{{Short description|Procedure for solving differential equations}}In [[mathematics]], '''variation of parameters''', also known as '''variation of constants''', is a general method to solve [[inhomogeneous differential equation|inhomogeneous]] [[linear differential equation|linear]] [[ordinary differential equation]]s.

For first-order inhomogeneous linear differential equations it is usually possible to find solutions via [[integrating factor]]s or [[method of undetermined coefficients|undetermined coefficients]] with considerably less effort, although those methods leverage [[heuristic]]s that involve guessing and don'tdo not work for all inhomogeneous linear differential equations.

Variation of parameters extends to linear [[partial differential equations]] as well, specifically to inhomogeneous problems for linear evolution equations like the [[heat equation]], [[wave equation]], and [[vibrating plate]] equation. In this setting, the method is more often known as [[Duhamel's principle]], named after [[Jean-Marie Duhamel]] (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa.