Variation of parameters: Difference between revisions

Revision as of 14:49, 17 March 2023

Latest revision as of 17:34, 13 August 2025

(3 intermediate revisions by 3 users not shown)

Line 1:

{{Differential equations}}

{{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 equationsequation]]s 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 do 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.

Line 36:

{{NumBlk|:|<math>y^{(n)}(x) + \sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = 0.</math>|{{EquationRef|ii}}}}

Then a [[ordinaryOrdinary differential equation#Solutions|particular solution]] to the non-homogeneous equation is given by

{{NumBlk|:|<math>y_p(x) = \sum_{i=1}^{n} c_i(x) y_i(x)</math>|{{EquationRef|iii}}}}

Line 271:

Note that <math>A(x)</math> and <math> B(x)</math> are each determined only up to an arbitrary additive constant (the [[constant of integration]]). Adding a constant to <math>A(x)</math> or <math>B(x)</math> does not change the value of <math>Lu_G(x)</math> because the extra term is just a linear combination of ''u''₁ and ''u''₂, which is a solution of <math>L</math> by definition.

==See also==

* [[Alekseev–Gröbner formula]], a generalization of the variation of constants formula.

* [[Reduction of order]]

== Notes ==

Line 307 ⟶ 311:

| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/

}}

==See also==

* [[Reduction of order]]

* [[Alekseev–Gröbner formula]], a generalization of the variation of constants formula.

== External links ==

*[http://tutorial.math.lamar.edu/classes/de/VariationofParameters.aspx Online Notes / Proof] by Paul Dawkins, [[Lamar University]].

*{{PlanetMath|VariationOfParameters}}

*[http://planetmath.org/encyclopedia/VariationOfParameters.html PlanetMath page].

*[https://projecteuclid.org/download/pdf_1/euclid.mjms/1316092232 A NOTE ON LAGRANGE’S METHOD OF VARIATION OF PARAMETERS]