Variation of parameters


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 don't 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 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, and later completed by the Italian-French mathematician Joseph-Louis Lagrange.
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.

Intuitive explanation

Consider the equation of the forced dispersionless spring, in suitable units:
Here is the displacement of the spring from the equilibrium, and is an external applied force that depends on time. When the external force is zero, this is the homogeneous equation.
We can construct the solution physically, as follows. Between times and, the momentum corresponding to the solution has a net change . A solution to the inhomogeneous equation, at the present time, is obtained by linearly superposing the solutions obtained in this manner, for going between 0 and.
The homogeneous initial-value problem, representing a small impulse being added to the solution at time, is
The unique solution to this problem is easily seen to be. The linear superposition of all of these solutions is given by the integral:
To verify that this satisfies the required equation:
as required.
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 Fds. Denote by the solution of the homogeneous initial value problem
Then a particular solution of the inhomogeneous equation is
the result of linearly superposing the infinitesimal homogeneous solutions. There are generalizations to higher order linear differential operators.
In practice, variation of parameters usually involves the fundamental solution of the homogeneous problem, the infinitesimal solutions then being given in terms of explicit linear combinations of linearly independent fundamental solutions. In the case of the forced dispersionless spring, the kernel is the associated decomposition into fundamental solutions.

Description of method

Given an ordinary non-homogeneous linear differential equation of order n
Let be a fundamental system 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 of n equations can then be solved using Cramer's rule yielding
where is the Wronskian determinant of the fundamental system and is the Wronskian determinant of the fundamental system with the i-th column replaced by
The particular solution to the non-homogeneous equation can then be written as

Examples

First-order equation

The general solution of the corresponding homogeneous equation is the complementary solution to our original equation:
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:
By substituting the particular solution into the non-homogeneous equation, we can find C:
We only need a single particular solution, so we arbitrarily select for simplicity. Therefore the particular solution is:
The final solution of the differential equation is:
This recreates the method of integrating factors.

Specific second-order equation

Let us solve
We want to find the general solution to the differential equation, that is, we want to find solutions to the homogeneous differential equation
The characteristic equation is:
Since is a repeated root, we have to introduce a factor of x for one solution to ensure linear independence: u1 = e−2x and u2 = xe−2x. 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.
We seek functions A and B so A'u1 + B'u2 is a particular solution of the non-homogeneous equation. We need only calculate the integrals
Recall that for this example
That is,
where and are constants of integration.

General second-order equation

We have a differential equation of the form
and we define the linear operator
where D represents the differential operator. We therefore have to solve the equation for, where and are known.
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 — call them u1 and u2 — we can proceed with variation of parameters.
Now, we seek the general solution to the differential equation which we assume to be of the form
Here, and are unknown and and are the solutions to the homogeneous 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
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 and B from these conditions, so, given
we can solve for, B′)T, so
where W denotes the Wronskian of u1 and u2. So,
While homogeneous equations are relatively easy to solve, this method allows the calculation of the coefficients of the general solution of the inhomogeneous equation, and thus the complete general solution of the inhomogeneous equation can be determined.
Note that and are each determined only up to an arbitrary additive constant. Adding a constant to or does not change the value of because the extra term is just a linear combination of u1 and u2, which is a solution of by definition.