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{{Differential equations}}
{{Short description|Procedure for solving differential equations}}In [[mathematics]], '''variation of parameters''', also known as '''variation of constants''', is a general method to solve [[inhomogeneous differential equation|inhomogeneous]]
For first-order inhomogeneous [[linear differential
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.
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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|Joseph-Louis Lagrange]] (1736–1813).<ref>See:
* [[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), [https://books.google.com/books?id=URPSrBntwdAC&pg=PA431
* Edgar Odell Lovett (1899) [https://books.google.com/books?id=j7sKAAAAIAAJ&pg=PA47
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.<ref>Euler, L. (1748) [https://books.google.com/books?id=GtA6Ea1NlqwC&pg=PA1
Lagrange first used the method in 1766.<ref>Lagrange, J.-L. (1766) [https://books.google.com/books?id=XwVNAAAAMAAJ&pg=RA1-PA179
* Lagrange, J.-L. (1781) [https://books.google.com/books?id=UitRAAAAYAAJ&pg=PA199
* Lagrange, J.-L. (1782) [https://books.google.com/books?id=kW9PAAAAYAAJ&pg=PA169
* Lagrange, J.-L. (1783) [https://books.google.com/books?id=Lz7fp3OnutEC&pg=PA161
* Lagrange, J.-L. (1778) [https://books.google.com/books?id=F90_AAAAYAAJ&pg=PA60-IA55
* Lagrange, J.-L. (1778) [https://books.google.com/books?id=F90_AAAAYAAJ&pg=PA60-IA68
* Lagrange, J.-L. (1783) [http://gallica.bnf.fr/ark:/12148/bpt6k229223s/f498.image "Sur le probleme de la détermination des orbites des cometes d'après trois observations. Troisième mémoire, dans lequel on donne une solution directe et générale du problème."], ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 296–332 [published in 1785].</ref> During 1808–1810, Lagrange gave the method of variation of parameters its final form in a third series of papers.<ref>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, [http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f715.image 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, [http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f773 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, [http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f811.image pages 809–816].</ref>
== Intuitive explanation ==▼
Consider the equation of the forced dispersionless spring, in suitable units:▼
:<math>x''(t) + x(t) = F(t).</math>▼
Here {{math|''x''}} is the displacement of the spring from the equilibrium {{math|''x'' {{=}} 0}}, and {{math|''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 <math>t=s</math> and <math>t=s+ds</math>, the momentum corresponding to the solution has a net change <math>F(s)\,ds</math> (see: [[Impulse (physics)]]). A solution to the inhomogeneous equation, at the present time {{math|''t'' > 0}}, is obtained by linearly superposing the solutions obtained in this manner, for {{math|''s''}} going between 0 and {{math|t}}.▼
The homogeneous initial-value problem, representing a small impulse <math>F(s)\,ds</math> being added to the solution at time <math>t=s</math>, is▼
:<math>x''(t)+x(t)=0,\quad x(s)=0,\ x'(s)=F(s)\,ds.</math>▼
The unique solution to this problem is easily seen to be <math>x(t) = F(s)\sin(t-s)\,ds</math>. The linear superposition of all of these solutions is given by the integral:▼
:<math>x(t) = \int_0^t F(s)\sin(t-s)\,ds.</math>▼
To verify that this satisfies the required equation:▼
:<math>x'(t)=\int_0^t F(s)\cos(t-s)\,ds</math>▼
:<math>x''(t) = F(t) - \int_0^tF(s)\sin(t-s)\,ds = F(t)-x(t),</math>▼
as required (see: [[Leibniz integral rule]]).▼
The general method of variation of parameters allows for solving an inhomogeneous linear equation▼
:<math>Lx(t)=F(t)</math>▼
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 <math>x_s </math> the solution of the homogeneous initial value problem ▼
:<math>Lx(t)=0, \quad x(s)=0,\ x'(s)=F (s)\,ds. </math>▼
Then a particular solution of the inhomogeneous equation is ▼
:<math>x (t)=\int_0^t x_s (t)\,ds,</math>▼
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 <math>x_s </math> then being given in terms of explicit linear combinations of linearly independent fundamental solutions. In the case of the forced dispersionless spring, the kernel <math>\sin(t-s)=\sin t\cos s - \sin s\cos t </math> is the associated decomposition into fundamental solutions.▼
== Description of method ==
Given an ordinary non-homogeneous [[linear differential equation]] of order ''n''
{{NumBlk|:|<math>y^{(n)}(x) + \sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = b(x).</math>|{{EquationRef|i}}}}
Let <math>y_1(x), \ldots, y_n(x)</math> be a [[
{{NumBlk|:|<math>y^{(n)}(x) + \sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = 0.</math>|{{EquationRef|ii}}}}
Then a [[
{{NumBlk|:|<math>y_p(x) = \sum_{i=1}^{n} c_i(x) y_i(x)</math>|{{EquationRef|iii}}}}
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{{NumBlk|:|<math>\sum_{i=1}^n c_i'(x) y_i^{(j)}(x) = 0, \quad j = 0,\ldots, n-2.</math>|{{EquationRef|iv}}}}
Starting with ({{EquationNote|iii}}), repeated differentiation combined with repeated use of ({{EquationNote|iv}}) gives
{{NumBlk|:|<math>y_p^{(j)}(x) = \sum_{i=1}^{n} c_i(x) y_i^{(j)}(x), \quad j=0,\ldots,n-1 \, .</math>|{{EquationRef|v}}}}
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{{NumBlk|:|<math>y_p^{(n)}(x)=\sum_{i=1}^n c_i'(x)y_i^{(n-1)}(x)+\sum_{i=1}^n c_i(x) y_i^{(n)}(x) \, .</math>|{{EquationRef|vi}}}}
By substituting ({{EquationNote|iii}}) into ({{EquationNote|i}}) and applying ({{EquationNote|v}}) and ({{EquationNote|vi}}) it follows that
{{NumBlk|:|<math>\sum_{i=1}^n c_i'(x) y_i^{(n-1)}(x) = b(x).</math>|{{EquationRef|vii}}}}
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:<math>c_i'(x) = \frac{W_i(x)}{W(x)}, \, \quad i=1,\ldots,n</math>
where <math>W(x)</math> is the [[Wronskian determinant]] of the
The particular solution to the non-homogeneous equation can then be written as
:<math>\sum_{i=1}^n y_i(x) \, \int \frac{W_i(x)}{W(x)}\, \mathrm dx.</math>
▲== Intuitive explanation ==
▲Consider the equation of the forced dispersionless spring, in suitable units:
▲:<math>x''(t) + x(t) = F(t).</math>
▲Here {{math|''x''}} is the displacement of the spring from the equilibrium {{math|''x'' {{=}} 0}}, and {{math|''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 <math>t=s</math> and <math>t=s+ds</math>, the momentum corresponding to the solution has a net change <math>F(s)\,ds</math> (see: [[Impulse (physics)]]). A solution to the inhomogeneous equation, at the present time {{math|''t'' > 0}}, is obtained by linearly superposing the solutions obtained in this manner, for {{math|''s''}} going between 0 and {{math|t}}.
▲The homogeneous initial-value problem, representing a small impulse <math>F(s)\,ds</math> being added to the solution at time <math>t=s</math>, is
▲:<math>x''(t)+x(t)=0,\quad x(s)=0,\ x'(s)=F(s)\,ds.</math>
▲The unique solution to this problem is easily seen to be <math>x(t) = F(s)\sin(t-s)\,ds</math>. The linear superposition of all of these solutions is given by the integral:
▲:<math>x(t) = \int_0^t F(s)\sin(t-s)\,ds.</math>
▲To verify that this satisfies the required equation:
▲:<math>x'(t)=\int_0^t F(s)\cos(t-s)\,ds</math>
▲:<math>x''(t) = F(t) - \int_0^tF(s)\sin(t-s)\,ds = F(t)-x(t),</math>
▲as required (see: [[Leibniz integral rule]]).
▲The general method of variation of parameters allows for solving an inhomogeneous linear equation
▲:<math>Lx(t)=F(t)</math>
▲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 <math>x_s </math> the solution of the homogeneous initial value problem
▲:<math>Lx(t)=0, \quad x(s)=0,\ x'(s)=F (s)\,ds. </math>
▲Then a particular solution of the inhomogeneous equation is
▲:<math>x (t)=\int_0^t x_s (t)\,ds,</math>
▲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 <math>x_s </math> then being given in terms of explicit linear combinations of linearly independent fundamental solutions. In the case of the forced dispersionless spring, the kernel <math>\sin(t-s)=\sin t\cos s - \sin s\cos t </math> is the associated decomposition into fundamental solutions.
== Examples ==
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=== First-order equation ===
:<math> y' + p(x)y = q(x) </math>
The
:
This homogeneous differential equation can be solved by different methods, for example [[separation of variables]]:
:<math>\frac{d}{dx} y + p(x)y = 0 </math>
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: <math>\lambda^2+4\lambda+4=(\lambda+2)^2=0 </math>
Since <math>\lambda=-2</math> is a repeated root, we have to introduce a factor of ''x'' for one solution to ensure linear independence:
: <math>W=\begin{vmatrix}
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We have the system of equations
:<math>\begin{
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x) \end{
\begin{
A'(x) \\
B'(x)\end{
\begin{bmatrix} 0 \\ f \end{bmatrix}.</math>
\begin{pmatrix}▼
Expanding,
:<math>\begin{
A'(x)u_1(x)+B'(x)u_2(x)\\ A'(x)u_1'(x)+B'(x)u_2'(x) \end{ = \begin{ So the above system determines precisely the conditions
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We seek ''A''(''x'') and ''B''(''x'') from these conditions, so, given
:<math>\begin{
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x)
\end{ \begin{
A'(x) \\
B'(x)\end{
\begin{
0\\
f\end{
we can solve for (''A''′(''x''), ''B''′(''x''))<sup>T</sup>, so
:<math>\begin{
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x)
\end{ \begin{
u_2'(x) & -u_2(x) \\
-u_1'(x) & u_1(x) \end{
\begin{
where ''W'' denotes the [[Wronskian]] of ''u''<sub>1</sub> and ''u''<sub>2</sub>. (We know that ''W'' is nonzero, from the assumption that ''u''<sub>1</sub> and ''u''<sub>2</sub> are linearly independent.) So,
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Note that <math>A(x)</math> and <math> B(x)</math> are each determined only up to an arbitrary additive constant (the [[constant of integration]]). Adding a constant to <math>A(x)</math> or <math>B(x)</math> does not change the value of <math>Lu_G(x)</math> because the extra term is just a linear combination of ''u''<sub>1</sub> and ''u''<sub>2</sub>, which is a solution of <math>L</math> by definition.
==See also==
* [[Alekseev–Gröbner formula]], a generalization of the variation of constants formula.
* [[Reduction of order]]
== Notes ==
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== External links ==
*[http://tutorial.math.lamar.edu/classes/de/VariationofParameters.aspx Online Notes / Proof] by Paul Dawkins, [[Lamar University]].
*{{PlanetMath|VariationOfParameters}}
*[https://projecteuclid.org/download/pdf_1/euclid.mjms/1316092232 A NOTE ON LAGRANGE’S METHOD OF VARIATION OF PARAMETERS]
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