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.
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.
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).
[1] 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.
[2] In his 1749 study of the motions of the earth, Euler obtained differential equations for the orbital elements.
[3] In 1753, he applied the method to his study of the motions of the moon.
[5] Between 1778 and 1783, he further developed the method in two series of memoirs: one on variations in the motions of the planets[6] and another on determining the orbit of a comet from three observations.
[7] During 1808–1810, Lagrange gave the method of variation of parameters its final form in a third series of papers.
[8] Given an ordinary non-homogeneous linear differential equation of order n Let
are differentiable functions which are assumed to satisfy the conditions Starting with (iii), repeated differentiation combined with repeated use of (iv) gives One last differentiation gives By substituting (iii) into (i) and applying (v) and (vi) it follows that The linear system (iv and vii) of n equations can then be solved using Cramer's rule yielding where
is the Wronskian determinant of the basis with the i-th column replaced by
The particular solution to the non-homogeneous equation can then be written as Consider the equation of the forced dispersionless spring, in suitable units: Here x is the displacement of the spring from the equilibrium x = 0, and F(t) is an external applied force that depends on time.
When the external force is zero, this is the homogeneous equation (whose solutions are linear combinations of sines and cosines, corresponding to the spring oscillating with constant total energy).
A solution to the inhomogeneous equation, at the present time t > 0, is obtained by linearly superposing the solutions obtained in this manner, for s going between 0 and t. The homogeneous initial-value problem, representing a small impulse
The linear superposition of all of these solutions is given by the integral: To verify that this satisfies the required equation: as required (see: Leibniz integral rule).
The general method of variation of parameters allows for solving an inhomogeneous linear equation by means of considering the second-order linear differential operator L to be the net force, thus the total impulse imparted to a solution between time s and s+ds is F(s)ds.
There are generalizations to higher order linear differential operators.
In the case of the forced dispersionless spring, the kernel
The complementary solution to our original (inhomogeneous) equation is the general solution of the corresponding homogeneous equation (written below): This homogeneous differential equation can be solved by different methods, for example separation of variables: The complementary solution to our original equation is therefore: Now we return to solving the non-homogeneous equation: Using the method variation of parameters, the particular solution is formed by multiplying the complementary solution by an unknown function C(x): By substituting the particular solution into the non-homogeneous equation, we can find C(x): We only need a single particular solution, so we arbitrarily select
is a repeated root, we have to introduce a factor of x for one solution to ensure linear independence:
The Wronskian of these two functions is Because the Wronskian is non-zero, the two functions are linearly independent, so this is in fact the general solution for the homogeneous differential equation (and not a mere subset of it).
We seek functions A(x) and B(x) so A(x)u1 + B(x)u2 is a particular solution of the non-homogeneous equation.
We must solve first the corresponding homogeneous equation: by the technique of our choice.
Once we've obtained two linearly independent solutions to this homogeneous differential equation (because this ODE is second-order) — call them u1 and u2 — we can proceed with variation of parameters.
Now, we seek the general solution to the differential equation
Since the above is only one equation and we have two unknown functions, it is reasonable to impose a second condition.
We choose the following: Now, Differentiating again (omitting intermediary steps) Now we can write the action of L upon uG as Since u1 and u2 are solutions, then We have the system of equations Expanding, So the above system determines precisely the conditions We seek A(x) and B(x) from these conditions, so, given we can solve for (A′(x), B′(x))T, so where W denotes the Wronskian of u1 and u2.
because the extra term is just a linear combination of u1 and u2, which is a solution of