Variation of parameters
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 (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.
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).
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 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 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:
The general method of variation of parameters allows for solution of an inhomogeneous linear equation
by interpreting the second-order linear differential operator L as the net force, so that the total impulse imparted to a solution between time s and s+ds is F(s)ds. 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.
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 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 (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 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
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 for simplicity. Therefore the particular solution is:
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 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 (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
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 (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 which we assume to be of the form
Here, and are unknown and and are the solutions to the homogeneous equation. (Observe that if and are constants, then .) 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:
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
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 and are each determined only up to an arbitrary additive constant (the constant of integration). 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.
- Forest Ray Moulton, An Introduction to Celestial Mechanics, 2nd ed. (first published by the Macmillan Company in 1914; reprinted in 1970 by Dover Publications, Inc., Mineola, New York), page 431.
- Edgar Odell Lovett (1899) "The theory of perturbations and Lie's theory of contact transformations," The Quarterly Journal of Pure and Applied Mathematics, vol. 30, pages 47–149; see especially pages 48–61.
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- Euler, L. (1749) "Recherches sur la précession des équinoxes, et sur la nutation de l’axe de la terre," Histoire [or Mémoires ] de l'Académie Royale des Sciences et Belles-lettres (Berlin), pages 289–325 [published in 1751].
- Euler, L. (1753) Theoria motus lunae: exhibens omnes ejus inaequalitates ... [The theory of the motion of the moon: demonstrating all of its inequalities ... ] (Saint Petersburg, Russia: Academia Imperialis Scientiarum Petropolitanae [Imperial Academy of Science (St. Petersburg)], 1753).
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- Lagrange, J.-L. (1783) "Théorie des variations périodiques des mouvemens des Planetes. Premiere partie, ... ," Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres (Berlin), pages 161–190.
- Lagrange, J.-L. (1778) "Sur le probleme de la détermination des orbites des cometes d'après trois observations, premier mémoire" (On the problem of determining the orbits of comets from three observations, first memoir), Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres (Berlin), pages 111–123 [published in 1780].
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