In
mathematics, variation of parameters, also known as variation of constants, is a general method to solve
inhomogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s.
For first-order inhomogeneous linear differential equations it is usually possible to find solutions via
integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calcul ...
s or
undetermined coefficients with considerably less effort, although those methods leverage
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
s that involve guessing and do not work for all inhomogeneous linear differential equations.
Variation of parameters extends to linear
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
as well, specifically to inhomogeneous problems for linear evolution equations like the
heat equation,
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
, and
vibrating plate equation. In this setting, the method is more often known as
Duhamel's principle In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to homogeneous differential equation, inhomogeneous linear evolution equations like the heat equation, wave equa ...
, named after
Jean-Marie Duhamel
Jean-Marie Constant Duhamel (; ; 5 February 1797 – 29 April 1872) was a French mathematician and physicist.
His studies were affected by the troubles of the Napoleonic era. He went on to form his own school ''École Sainte-Barbe''. Duhame ...
(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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(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.
[See:
* Lagrange, J.-L. (1808) “Sur la théorie des variations des éléments des planètes et en particulier des variations des grands axes de leurs orbites,” ''Mémoires de la première Classe de l’Institut de France''. Reprinted in: Joseph-Louis Lagrange with Joseph-Alfred Serret, ed., ''Oeuvres de Lagrange'' (Paris, France: Gauthier-Villars, 1873), vol. 6]
pages 713–768
* Lagrange, J.-L. (1809) “Sur la théorie générale de la variation des constantes arbitraires dans tous les problèmes de la méchanique,” ''Mémoires de la première Classe de l’Institut de France''. Reprinted in: Joseph-Louis Lagrange with Joseph-Alfred Serret, ed., ''Oeuvres de Lagrange'' (Paris, France: Gauthier-Villars, 1873), vol. 6
pages 771–805
* Lagrange, J.-L. (1810) “Second mémoire sur la théorie générale de la variation des constantes arbitraires dans tous les problèmes de la méchanique, ... ,” ''Mémoires de la première Classe de l’Institut de France''. Reprinted in: Joseph-Louis Lagrange with Joseph-Alfred Serret, ed., ''Oeuvres de Lagrange'' (Paris, France: Gauthier-Villars, 1873), vol. 6
pages 809–816
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 ( and ) 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
:
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 (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
(see:
Impulse (physics)). 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 (see:
Leibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integral are ).
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''. 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.
Examples
First-order equation
:
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
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
:
:
:
:
:
:
:
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:
:
This recreates the method of
integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calcul ...
s.
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:
and
. The
Wronskian
In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.
Definition
The Wronskian o ...
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'')''u''
1 + ''B''(''x'')''u''
2 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 (because this ODE is second-order) — call them ''u''
1 and ''u''
2 — 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:
:
Now,
:
Differentiating again (omitting intermediary steps)
:
Now we can write the action of ''L'' upon ''u''
''G'' as
:
Since ''u''
1 and ''u''
2 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
In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.
Definition
The Wronskian o ...
of ''u''
1 and ''u''
2. (We know that ''W'' is nonzero, from the assumption that ''u''
1 and ''u''
2 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 ''in''homogeneous 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
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
). Adding a constant to
or
does not change the value of
because the extra term is just a linear combination of ''u''
1 and ''u''
2, which is a solution of
by definition.
Notes
References
*
*
* {{cite book
, last = Teschl
, first = Gerald
, author-link = Gerald Teschl
, title = Ordinary Differential Equations and Dynamical Systems
, publisher =
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, year = 2012
, url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/
See also
*
Reduction of order
Reduction of order is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y_1(x) is known and a second linearly independent solution y_2(x) is desired. The method also appl ...
*
Alekseev–Gröbner formula, a generalization of the variation of constants formula.
External links
Online Notes / Proofby Paul Dawkins,
Lamar University
Lamar University (Lamar or LU) is a public university in Beaumont, Texas. Lamar has been a member of the Texas State University System since 1995. It was the flagship institution of the former Lamar University System. As of the fall of 2021, t ...
.
PlanetMath pageA NOTE ON LAGRANGE’S METHOD OF VARIATION OF PARAMETERS
Ordinary differential equations