In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
Contents
- History
- Description of method
- First order equation
- Specific second order equation
- General second order equation
- References
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 introduced by the Swiss-born mathematician Leonhard Euler (1707–1783) and 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; and 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, Lagrange further developed the method both in a series of memoirs on variations in the motions of the planets and in another series of memoirs on determining the orbit of a comet from three observations. (It should be noted that Euler and Lagrange applied this method to nonlinear differential equations and that, instead of varying the coefficients of linear combinations of solutions to homogeneous equations, they varied the constants of the unperturbed motions of the celestial bodies.) During 1808-1810, Lagrange gave the method of variation of parameters its final form in a series of papers. The central result of his study was the system of planetary equations in the form of Lagrange, which described the evolution of the Keplerian parameters (orbital elements) of a perturbed orbit.
In his description of evolving orbits, Lagrange set a reduced two-body problem as an unperturbed solution, and presumed that all perturbations come from the gravitational pull which the bodies other than the primary exert at the secondary (orbiting) body. Accordingly, his method implied that the perturbations depend solely on the position of the secondary, but not on its velocity. In the 20th century, celestial mechanics began to consider interactions which depend on both positions and velocities (relativistic corrections, atmospheric drag, inertial forces). Therefore, the method of variation of parameters used by Lagrange was extended to the situation with velocity-dependent forces.
Description of method
Given an ordinary non-homogeneous linear differential equation of order n
let
Then a particular solution to the non-homogeneous equation is given by
where the
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
The particular solution to the non-homogeneous equation can then be written as
First order equation
The general solution of the corresponding homogeneous equation (written below) is the complementary solution to our original (inhomogeneous) 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(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
The final solution of the differential equation is:
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
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 general solution of the non-homogeneous equation. We need only calculate the integrals
Recall that for this example
That is,
where
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
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
Here,
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. (We know that W is nonzero, from the assumption that u1 and u2 are linearly independent.) 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