In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics 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. History The method of variation of parameters was first sketched by the Swiss mathematician Leonhard Euler (1707–1783), and later completed by the Italian-French mathematician Joseph-Louis Lagrange (1736–1813). A forerunner of the method of variation of a celestial body's orbital elements appeared in Euler's work in 1748, while he was studying the mutual perturbations of Jupiter and Saturn. In his 1749 study of the motions of the earth, Euler obtained differential equations for the orbital elements. In 1753, he applied the method to his study of the motions of the moon. Lagrange first used the method in 1766. Between 1778 and 1783, he further developed the method in two series of memoirs: one on variations in the motions of the planets and another on determining the orbit of a comet from three observations. During 1808–1810, Lagrange gave the method of variation of parameters its final form in a third series of papers. Description of method Given an ordinary non-homogeneous linear differential equation of order n Let be a basis of the vector space of solutions of the corresponding homogeneous equation Then a particular solution to the non-homogeneous equation is given by where the are differentiable functions which are assumed to satisfy the conditions Starting with (), repeated differentiation combined with repeated use of () gives One last differentiation gives By substituting () into () and applying () and () it follows that The linear system ( and ) of n equations can then be solved using Cramer's rule yielding where is the Wronskian determinant of the basis and is the Wronskian determinant of the basis with the i-th column