Variation of parameters: Difference between revisions

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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]] [[linear differential equation|linear]] [[ordinary differential equation]]s.

For first-order inhomogeneous [[linear differential equationsequation]]s it is usually possible to find solutions via [[integrating factor]]s or [[method of undetermined coefficients|undetermined coefficients]] with considerably less effort, although those methods leverage [[heuristic]]s that involve guessing and don'tdo not 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.

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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#v=onepage&q&f=false page 431].

* Edgar Odell Lovett (1899) [https://books.google.com/books?id=j7sKAAAAIAAJ&pg=PA47#v=onepage&q&f=false "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.</ref>

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#v=onepage&q&f=false "Recherches sur la question des inégalités du mouvement de Saturne et de Jupiter, sujet proposé pour le prix de l'année 1748, par l’Académie Royale des Sciences de Paris"] [Investigations on the question of the differences in the movement of Saturn and Jupiter; this subject proposed for the prize of 1748 by the Royal Academy of Sciences (Paris)] (Paris, France: G. Martin, J.B. Coignard, & H.L. Guerin, 1749).</ref> In his 1749 study of the motions of the earth, Euler obtained differential equations for the orbital elements.<ref>Euler, L. (1749) [https://books.google.com/books?id=xA0_AAAAYAAJ&pg=PA289#v=onepage&q&f=false "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].</ref> In 1753, he applied the method to his study of the motions of the moon.<ref>Euler, L. (1753) [https://archive.org/details/theoriamotuslun00eulegoog 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).</ref>

Lagrange first used the method in 1766.<ref>Lagrange, J.-L. (1766) [https://books.google.com/books?id=XwVNAAAAMAAJ&pg=RA1-PA179#v=onepage&q&f=false “Solution de différens problèmes du calcul integral,”] ''Mélanges de philosophie et de mathématique de la Société royale de Turin'', vol. 3, pages 179–380.</ref> Between 1778 and 1783, he further developed the method in two series of memoirs: one on variations in the motions of the planets<ref>See:

* Lagrange, J.-L. (1781) [https://books.google.com/books?id=UitRAAAAYAAJ&pg=PA199#v=onepage&q&f=false "Théorie des variations séculaires des élémens des Planetes. Premiere partie, ... ,"] ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 199–276.

* Lagrange, J.-L. (1782) [https://books.google.com/books?id=kW9PAAAAYAAJ&pg=PA169#v=onepage&q&f=false "Théorie des variations séculaires des élémens des Planetes. Seconde partie, ... ,"] ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 169–292.

* Lagrange, J.-L. (1783) [https://books.google.com/books?id=Lz7fp3OnutEC&pg=PA161#v=onepage&q&f=false "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.</ref> and another on determining the orbit of a comet from three observations.<ref>See:

* Lagrange, J.-L. (1778) [https://books.google.com/books?id=F90_AAAAYAAJ&pg=PA60-IA55#v=onepage&q&f=false "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].

* Lagrange, J.-L. (1778) [https://books.google.com/books?id=F90_AAAAYAAJ&pg=PA60-IA68#v=onepage&q&f=false "Sur le probleme de la détermination des orbites des cometes d'après trois observations, second mémoire"], ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-lettres'' (Berlin), pages 124–161 [published in 1780].

* 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>

== 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).\quad\quad {\rm (i)}</math>|{{EquationRef|i}}}}

Let <math>y_1(x), \ldots, y_n(x)</math> be a [[FundamentalBasis system#Homogeneous(linear equationsalgebra)|fundamentalbasis]] systemof the [[vector space]] of solutions of the corresponding homogeneous equation

{{NumBlk|:|<math>y^{(n)}(x) + \sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = 0.\quad\quad {\rm (ii)}</math>|{{EquationRef|ii}}}}

Then a [[ordinaryOrdinary differential equation#Solutions|particular solution]] to the non-homogeneous equation is given by

{{NumBlk|:|<math>y_p(x) = \sum_{i=1}^{n} c_i(x) y_i(x)\quad\quad {\rm (iii)}</math>|{{EquationRef|iii}}}}

where the <math>c_i(x)</math> are differentiable functions which are assumed to satisfy the conditions

{{NumBlk|:|<math>\sum_{i=1}^n c_i'(x) y_i^{(j)}(x) = 0, \quad j = 0,\ldots, n-2.\quad\quad {\rm (iv)}</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}}}}

One last differentiation gives

{{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) \, \mathrm{.</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).\quad\quad {\rm (vii)}</math>|{{EquationRef|vii}}}}

The linear system ({{EquationNote|iv}} and {{EquationNote|vii}}) of ''n'' equations can then be solved using [[Cramer's rule]] yielding

:<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 fundamentalbasis system<math>y_1(x), \ldots, y_n(x)</math> and <math>W_i(x)</math> is the Wronskian determinant of the fundamental systembasis with the ''i''-th column replaced by <math>(0, 0, \ldots, b(x)).</math>

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 ==

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:<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>

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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''

:<math>y^{(n)}(x) + \sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = b(x).\quad\quad {\rm (i)}</math>

Let <math>y_1(x), \ldots, y_n(x)</math> be a [[Fundamental system#Homogeneous equations|fundamental system]] of solutions of the corresponding homogeneous equation

:<math>y^{(n)}(x) + \sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = 0.\quad\quad {\rm (ii)}</math>

Then a [[ordinary differential equation|particular solution]] to the non-homogeneous equation is given by

:<math>y_p(x) = \sum_{i=1}^{n} c_i(x) y_i(x)\quad\quad {\rm (iii)}</math>

where the <math>c_i(x)</math> are differentiable functions which are assumed to satisfy the conditions

:<math>\sum_{i=1}^n c_i'(x) y_i^{(j)}(x) = 0, \quad j = 0,\ldots, n-2.\quad\quad {\rm (iv)}</math>

Starting with (iii), repeated differentiation combined with repeated use of (iv) gives

:<math>y_p^{(j)}(x) = \sum_{i=1}^{n} c_i(x) y_i^{(j)}(x),

\quad j=0,\ldots,n-1 \, \mathrm{.} \quad\quad {\rm (v)}</math>

One last differentiation gives

:<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)\, \mathrm{.}

\quad\quad{\rm (vi)}</math>

By substituting (iii) into (i) and applying (v) and (vi) it follows that

:<math>\sum_{i=1}^n c_i'(x) y_i^{(n-1)}(x) = b(x).\quad\quad {\rm (vii)}</math>

The linear system (iv and vii) of ''n'' equations can then be solved using [[Cramer's rule]] yielding

:<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 fundamental system and <math>W_i(x)</math> is the Wronskian determinant of the fundamental system with the ''i''-th column replaced by <math>(0, 0, \ldots, b(x)).</math>

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>

== Examples ==

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=== First-order equation ===

:<math> y' + p(x)y = q(x) </math>

The generalcomplementary solution ofto theour corresponding homogeneous equationoriginal (written belowinhomogeneous) equation is the complementarygeneral solution toof ourthe originalcorresponding (inhomogeneous)homogeneous equation (written below):

: <math> y' + p(x)y = 0 </math>.

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>

Line 105 ⟶ 103:

:<math>\frac{dy}{dx}=-p(x)y </math>

:<math>{dy \over y} = -{p(x)\,dx},</math>

:<math>\int \frac{1}{ y} \, dy = -\int p(x) \, dx </math>

Line 116 ⟶ 114:

Now we return to solving the non-homogeneous equation:

: <math> y' + p(x)y = q(x)</math>

Using the method variation of parameters, the particular solution is formed by multiplying the complementary solution by an unknown function ''C''(''x''):

:<math>y_p = C(x) e^{-\int p(x) \, dx}</math>

By substituting the particular solution into the non-homogeneous equation, we can find ''C''(''x''):

: <math> C' (x) e^{-\int p(x) \, dx} - C(x) p(x) e^{-\int p(x) \, dx} + p(x) C(x) e^{-\int p(x) \, dx} = q(x)</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: ''u''<submath>1</sub>&nbsp; u_1 =&nbsp;'' e''<sup>−2''x''^{-2x} </supmath> and ''u''<submath>2</sub>&nbsp; u_2 =&nbsp;''xe''<sup>−2''x'' e^{-2x}</supmath>. The [[Wronskian]] of these two functions is

: <math>W=\begin{vmatrix}

Line 164 ⟶ 162:

That is,

:<math>A(x) = - \int {1\over e^{-4x}} xe^{-2x} \cosh x \,\mathrm dx = - \int xe^{2x}\cosh x \,\mathrm dx = -{1\over 18}e^x\left(9(x-1)+e^{2x}(3x-1)\right)+C_1</math>

:<math>B(x) = \int {1 \over e^{-4x}} e^{-2x} \cosh x \,\mathrm dx = \int e^{2x}\cosh x\,\mathrm dx ={1\over 6}e^{x}\left(3+e^{2x}\right)+C_2 </math>

where <math>C_1</math> and <math>C_2</math> are constants of integration.

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We have the system of equations

:<math>\begin{pmatrixbmatrix}

u_1(x) & u_2(x) \\

u_1'(x) & u_2'(x) \end{pmatrixbmatrix}

\begin{pmatrixbmatrix}

A'(x) \\

B'(x)\end{pmatrixbmatrix} =

\begin{bmatrix} 0 \\ f \end{bmatrix}.</math>

\begin{pmatrix}

0\\

f\end{pmatrix}.</math>

Expanding,

:<math>\begin{pmatrixbmatrix}
A'(x)u_1(x)+B'(x)u_2(x)\\
A'(x)u_1'(x)+B'(x)u_2'(x) \end{pmatrixbmatrix}
= \begin{pmatrixbmatrix} 0\\f\end{pmatrixbmatrix}.</math>

So the above system determines precisely the conditions

Line 239 ⟶ 238:

We seek ''A''(''x'') and ''B''(''x'') from these conditions, so, given

:<math>\begin{pmatrixbmatrix}

u_1(x) & u_2(x) \\

u_1'(x) & u_2'(x)
\end{pmatrixbmatrix}

\begin{pmatrixbmatrix}

A'(x) \\

B'(x)\end{pmatrixbmatrix} =

\begin{pmatrixbmatrix}

0\\

f\end{pmatrixbmatrix}</math>

we can solve for (''A''′(''x''), ''B''′(''x''))^''T'', so

:<math>\begin{pmatrixbmatrix} A'(x) \\ B'(x) \end{bmatrix} =

\begin{pmatrixbmatrix}

A'(x) \\

B'(x)\end{pmatrix}=

\begin{pmatrix}

u_1(x) & u_2(x) \\

u_1'(x) & u_2'(x)
\end{pmatrixbmatrix}^{-1}

\begin{pmatrixbmatrix} 0\\ f \end{pmatrixbmatrix} =\frac{1}{W} \begin{pmatrixbmatrix}

u_2'(x) & -u_2(x) \\

-u_1'(x) & u_1(x) \end{pmatrixbmatrix}

\begin{pmatrixbmatrix} 0\\ f \end{pmatrixbmatrix},</math>

where ''W'' denotes the [[Wronskian]] 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,

:<math> \begin{align}

A'(x) &= - {1\over W} u_2(x) f(x),\; & B'(x) &= {1 \over W} u_1(x)f(x) \\

A(x) &= - \int {1\over W} u_2(x) f(x)\,\mathrm dx,\; & B(x) &= \int {1 \over W} u_1(x)f(x)\,\mathrm dx

\end{align}</math>

Line 272 ⟶ 271:

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''₁ and ''u''₂, 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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| publisher = [[American Mathematical Society]]

| year = 2012

| url = httphttps://www.mat.univie.ac.at/~gerald/ftp/book-ode/

}}

== External links ==

*[http://tutorial.math.lamar.edu/classes/de/VariationofParameters.aspx Online Notes / Proof] by Paul Dawkins, [[Lamar University]].

*{{PlanetMath|VariationOfParameters}}

*[http://planetmath.org/encyclopedia/VariationOfParameters.html PlanetMath page].

*[https://projecteuclid.org/download/pdf_1/euclid.mjms/1316092232 A NOTE ON LAGRANGE’S METHOD OF VARIATION OF PARAMETERS]