Variation of parameters

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, variation of parameters, also known as variation of constants, is a general method to solve <a href="/facts/Inhomogeneous_differential_equation/ygKGQ8kD">inhomogeneous</a> linear <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a>.
For first-order inhomogeneous <a href="/facts/Linear_differential_equation/nLEbIIVf">linear differential equations</a> it is usually possible to find solutions via <a href="/facts/Integrating_factor/5tKufbqg">integrating factors</a> or <a href="/facts/Method_of_undetermined_coefficients/L4jKFpPE">undetermined coefficients</a> with considerably less effort, although those methods leverage <a href="/facts/Heuristic/BcBEkv9c">heuristics</a> that involve guessing and do not work for all inhomogeneous linear differential equations.
Variation of parameters extends to linear <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a> as well, specifically to inhomogeneous problems for linear evolution equations like the <a href="/facts/Heat_equation/2JPPr5GG">heat equation</a>, <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a>, and <a href="/facts/Vibrating_plate/xULTrFH8">vibrating plate</a> equation. In this setting, the method is more often known as <a href="/facts/Duhamel%27s_principle/1plo3SyL">Duhamel's principle</a>, named after <a href="/facts/Jean-Marie_Duhamel/Rb8q3HWg">Jean-Marie Duhamel</a> (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.

Variation of parameters open-in-new

Variation of parameters