Content deleted Content added
Tags: Mobile edit Mobile web edit |
m →External links: fix dead link |
||
| (43 intermediate revisions by 30 users not shown) | |||
Line 1:
{{Differential equations}}
For first-order inhomogeneous [[linear differential equation]]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 do 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.
== History ==
The method of variation of parameters was
* [[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 "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 "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 "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 “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 "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 "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 "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 "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 "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).</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.
Then a [[
{{NumBlk|:|<math>y_p(x) = \sum_{i=1}^{n} c_i(x) y_i(x)
where the <math>c_i(x)</math> are differentiable functions which are assumed to satisfy the conditions
{{NumBlk|:|<math>\sum_{i=1}^
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 \
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) \,
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).
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
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 ==
=== First
:<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>
Line 102 ⟶ 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>
:<math>\ln |y| = -\int p(x) \, dx +
:<math>y = \pm e^{-\int p(x) \, dx +
The complementary solution to our original equation is therefore:
:<math>y_c = C_0 e^{-\int p(x) \, dx}</math>
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>
Line 126 ⟶ 127:
:<math>y_p =e^{-\int p(x) \, dx} \int q(x) e^{\int p(x) \, dx} \, dx</math>
The final solution of the differential equation is:
:<math>
y &= y_c + y_p\\
&=C_0 e^{-\int p(x) \, dx} + e^{-\int p(x) \, dx} \int q(x) e^{\int p(x) \, dx} \, dx
\end{align}</math>
This recreates the method of [[integrating factor]]s.
=== Specific second
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
Line 139 ⟶ 143:
: <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}
Line 148 ⟶ 152:
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''<sub>1</sub> + ''B''(''x'')''u''<sub>2</sub> is a
:<math>A(x) = - \int {1\over W} u_2(x) b(x)\,\mathrm dx,\; B(x) = \int {1 \over W} u_1(x)b(x)\,\mathrm dx</math>
Line 154 ⟶ 158:
Recall that for this example
:<math>b(x) = \cosh
That is,
:<math>A(x) = - \int {1\over e^{-4x}} xe^{-2x} \cosh
:<math>B(x) = \int {1 \over e^{-4x}} e^{-2x} \cosh
where <math>C_1</math> and <math>C_2</math> are constants of integration.
=== General second
We have a differential equation of the form
Line 212 ⟶ 216:
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>
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
Line 233 ⟶ 238:
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>
:<math>\begin{
\begin{bmatrix}
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,
:<math> \begin{align}
A'(x) &= - {1\over W} u_2(x) f(x),
A(x) &= - \int {1\over W} u_2(x) f(x)\,\mathrm dx,
\end{align}</math>
Line 267 ⟶ 272:
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
* [[Alekseev–Gröbner formula]], a generalization of the variation of constants formula.
* [[Reduction of order]]
== Notes ==
{{reflist}}
== References ==
* {{cite book
| last1 = Coddington
| first1 = Earl A.
| last2 = Levinson
| first2 = Norman
| title = Theory of Ordinary Differential Equations
| url = https://archive.org/details/theoryofordinary00codd
| url-access = registration
| publisher = [[McGraw-Hill]]
| year = 1955
}}
* {{cite book
| first1 = William E.
| last1 = Boyce
| first2 = Richard C.
| last2 = DiPrima
| title = Elementary Differential Equations and Boundary Value Problems
| edition = 8th
| publisher = Wiley
| year = 2005
| pages = 186–192, 237–241
}}
* {{cite book
|
|
|
| title = Ordinary Differential Equations and Dynamical Systems
| publisher = [[American Mathematical Society]]
| year = 2012
| url = https://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}}
*[https://projecteuclid.org/download/pdf_1/euclid.mjms/1316092232 A NOTE ON LAGRANGE’S METHOD OF VARIATION OF PARAMETERS]
| |||