|
|
| Line 1: |
Line 1: |
| 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.
| | Whenever a association struggle begins, you will see Often the particular War Map, a good map of this combat area area association competitions booty place. Useful territories will consistently be on the left, thanks to the adversary association on the inside the right. Any boondocks anteroom on the war map represents some kind of war base.<br><br>Video games are fun to use your kids. This can help you learn much more details about your kid's interests. Sharing interests with your kids like this can often create great conversations. It also gives you an opportunity to monitor developing on their skills.<br><br>Video games are very well-liked in many homes. When you loved this article and you wish to receive more information with regards to clash of clans hack cydia ([http://prometeu.net continue reading this]) please visit our page. The associated with people perform online online video media to pass through time, however, some blessed individuals are paid to experience clash of clans sur pc. Casino is going to automatically be preferred for some work-time into the future. These tips will help you if you are planning to try out online.<br><br>On line [http://www.tumblr.com/tagged/games+acquire games acquire] more to offer your son aka daughter than only a possibility to capture points. Try deciding on adventure titles that instruct your young person some thing. Basically an example, sports exercises video games will assist your youngster learn any guidelines for game titles, and exactly how web based games are played finally out. Check out some testimonials in discover game titles that supply a learning experience instead of just mindless, repeated motion.<br><br>Home pc games are a lot of fun, but individuals could be very tricky, also. If buyers are put on a brand new game, go on generally web and also appear for cheats. Most games have some model of cheat or secrets-and-cheats that can make him or her a lot easier. Only search in your own favorite search engine and even you can certainly discover cheats to get this action better.<br><br>This kind of information, we're accessible to assist you to alpha dog substituting offers. Application Clash of Clans Cheats' data, let's say suitable for archetype you appetite 1hr (3, 600 seconds) in order to bulk 20 gems, and additionally 1 day (90, 4000 seconds) to help bulk 260 gems. Similar to appropriately stipulate a operation for this kind about band segment.<br><br>To assist you to conclude, clash of clans hack tool no review must not be able to get in approach of the bigger question: what makes we here? [http://www.alexa.com/search?q=Putting&r=topsites_index&p=bigtop Putting] this aside the truck bed cover's of great importance. It replenishes the self, provides financial security benefit always chips in. |
| | |
| For first-order inhomogeneous linear differential equations 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't work for all inhomogenous 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]] 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 introduced by the Swiss-born mathematician [[Leonhard Euler]] (1707–1783) and 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), [http://books.google.com/books?id=URPSrBntwdAC&pg=PA431#v=onepage&q&f=false page 431].
| |
| * Edgar Odell Lovett (1899) [http://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) [http://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) [http://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> and in 1753 he applied the method to his study of the motions of the moon.<ref>Euler, L. (1753) [http://books.google.com/books?id=2c9NAAAAMAAJ&pg=PR4#v=onepage&q&f=false 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) [http://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, Lagrange further developed the method both in a series of memoirs on variations in the motions of the planets<ref>See:
| |
| * Lagrange, J.-L. (1781) [http://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) [http://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) [http://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 in another series of memoirs on determining the orbit of a comet from three observations.<ref>See:
| |
| * Lagrange, J.-L. (1778) [http://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) [http://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> (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.<ref>Michael Efroimsky (2002) [http://arxiv.org/pdf/astro-ph/0212245.pdf "Implicit gauge symmetry emerging in the N-body problem of celestial mechanics,"] page 3.</ref>) During 1808-1810, Lagrange gave the method of variation of parameters its final form in a 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> 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.<ref>See:
| |
| * Michael Efroimsky (2005) [http://onlinelibrary.wiley.com/doi/10.1196/annals.1370.016/abstract "Gauge Freedom in Orbital Mechanics." ANYAS, Vol. 1065, pp. 346–374 (2005)]
| |
| * Michael Efroimsky and Peter Goldreich (2004) [http://www.aanda.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/aa/abs/2004/09/aa0058/aa0058.html "Gauge symmetry of the N-body problem of Celestial Mechanics." Astronomy and Astrophysics, Vol. 415, pp. 1187–1199. (2004)]
| |
| * Michael Efroimsky and Peter Goldreich (2003) [http://scitation.aip.org/content/aip/journal/jmp/44/12/10.1063/1.1622447 "Gauge symmetry of the N-body problem in the Hamilton-Jacobi approach." Journal of Mathematical Physics, Vol. 44, pp. 5958–5977. (2003)]</ref>
| |
| | |
| ==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 \, \mathrm{,} \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) \, \mathrm{,}
| |
| \quad j=0,\ldots,n-1 \, \mathrm{.} \quad\quad {\rm (v)}</math> <br>
| |
| | |
| 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><br>
| |
| | |
| 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> <br>
| |
| | |
| 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)} dx.</math>
| |
| | |
| == Examples ==
| |
| === Specific second order equation ===
| |
| Let us solve
| |
| : <math> y''+4y'+4y=\cosh{x}.\;\!</math>
| |
| | |
| We want to find the general solution to the differential equation, that is, we want to find solutions to the homogeneous differential equation
| |
| : <math>y''+4y'+4y=0.\;\!</math>
| |
| From the characteristic equation
| |
| : <math>\lambda^2+4\lambda+4=(\lambda+2)^2=0\;\!</math>
| |
| : <math>\lambda=-2.\;\!</math>
| |
| Since we have a repeated root, we have to introduce a factor of ''x'' for one solution to ensure linear independence.
| |
| | |
| So, we obtain ''u''<sub>1</sub> = ''e''<sup>−2''x''</sup>, and ''u''<sub>2</sub> = ''xe''<sup>−2''x''</sup>. The [[Wronskian]] of these two functions is
| |
| : <math>\begin{vmatrix}
| |
| e^{-2x} & xe^{-2x} \\
| |
| -2e^{-2x} & -e^{-2x}(2x-1)\\
| |
| \end{vmatrix} = -e^{-2x}e^{-2x}(2x-1)+2xe^{-2x}e^{-2x} </math>
| |
| :<math>= -e^{-4x}(2x-1)+2xe^{-4x}= (-2x+1+2x)e^{-4x} = e^{-4x}.\;\!</math>
| |
| | |
| 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 general solution of the non-homogeneous equation. We need only calculate the integrals
| |
| :<math>A(x) = - \int {1\over W} u_2(x) b(x)\,dx,\; B(x) = \int {1 \over W} u_1(x)b(x)\,dx</math>
| |
| that is, | |
| :<math>A(x) = - \int {1\over e^{-4x}} xe^{-2x} \cosh{x}\,dx = - \int xe^{2x}\cosh{x}\,dx = -{1\over 18}e^x(9(x-1)+e^{2x}(3x-1))+C_1</math>
| |
| :<math>B(x) = \int {1 \over e^{-4x}} e^{-2x} \cosh{x}\,dx = \int e^{2x}\cosh{x}\,dx ={1\over 6}e^{x}(3+e^{2x})+C_2 </math>
| |
| where <math>C_1</math> and <math>C_2</math> are constants of integration.
| |
| | |
| === General second order equation ===
| |
| We have a differential equation of the form
| |
| :<math>u''+p(x)u'+q(x)u=f(x)\,</math>
| |
| and we define the linear operator
| |
| :<math>L=D^2+p(x)D+q(x)\,</math>
| |
| where ''D'' represents the [[differential operator]]. We therefore have to solve the equation <math>L u(x)=f(x)</math> for <math>u(x)</math>, where <math>L</math> and <math>f(x)</math> are known.
| |
| | |
| We must solve first the corresponding homogeneous equation:
| |
| :<math>u''+p(x)u'+q(x)u=0\,</math>
| |
| 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''<sub>1</sub> and ''u''<sub>2</sub> — we can proceed with variation of parameters.
| |
| | |
| Now, we seek the general solution to the differential equation <math> u_G(x)</math> which we assume to be of the form
| |
| :<math>u_G(x)=A(x)u_1(x)+B(x)u_2(x).\,</math>
| |
| | |
| Here, <math>A(x)</math> and <math>B(x)</math> are unknown and <math>u_1(x)</math> and <math>u_2(x)</math> are the solutions to the homogeneous equation. Observe that if <math>A(x)</math> and <math>B(x)</math> are constants, then <math>Lu_G(x)=0</math>. We desire ''A''=''A''(''x'') and ''B''=''B''(''x'') to be of the form
| |
| :<math>A'(x)u_1(x)+B'(x)u_2(x)=0.\,</math>
| |
| | |
| Now,
| |
| :<math>u_G'(x)=(A(x)u_1(x)+B(x)u_2(x))'=(A(x)u_1(x))'+(B(x)u_2(x))'\,</math>
| |
| :<math>=A'(x)u_1(x)+A(x)u_1'(x)+B'(x)u_2(x)+B(x)u_2'(x)\,</math>
| |
| :<math>=A'(x)u_1(x)+B'(x)u_2(x)+A(x)u_1'(x)+B(x)u_2'(x)\,</math>
| |
| and since we have required the above condition, then we have | |
| :<math>u_G'(x)=A(x)u_1'(x)+B(x)u_2'(x).\,</math>
| |
| Differentiating again (omitting intermediary steps)
| |
| :<math>u_G''(x)=A(x)u_1''(x)+B(x)u_2''(x)+A'(x)u_1'(x)+B'(x)u_2'(x).\,</math>
| |
| | |
| Now we can write the action of ''L'' upon ''u''<sub>''G''</sub> as
| |
| :<math>Lu_G=A(x)Lu_1(x)+B(x)Lu_2(x)+A'(x)u_1'(x)+B'(x)u_2'(x).\,</math>
| |
| Since ''u''<sub>1</sub> and ''u''<sub>2</sub> are solutions, then
| |
| :<math>Lu_G=A'(x)u_1'(x)+B'(x)u_2'(x).\,</math>
| |
| | |
| We have the system of equations
| |
| :<math>\begin{pmatrix}
| |
| u_1(x) & u_2(x) \\
| |
| u_1'(x) & u_2'(x) \end{pmatrix}
| |
| \begin{pmatrix}
| |
| A'(x) \\
| |
| B'(x)\end{pmatrix} =
| |
| \begin{pmatrix}
| |
| 0\\
| |
| f\end{pmatrix}.</math>
| |
| Expanding,
| |
| :<math>\begin{pmatrix}
| |
| A'(x)u_1(x)+B'(x)u_2(x)\\
| |
| A'(x)u_1'(x)+B'(x)u_2'(x)\end{pmatrix} =
| |
| \begin{pmatrix}
| |
| 0\\f\end{pmatrix}.</math>
| |
| So the above system determines precisely the conditions
| |
| :<math>A'(x)u_1(x)+B'(x)u_2(x)=0\,</math>
| |
| :<math>A'(x)u_1'(x)+B'(x)u_2'(x)=Lu_G=f.\,</math>
| |
| | |
| We seek ''A''(''x'') and ''B''(''x'') from these conditions, so, given
| |
| :<math>\begin{pmatrix}
| |
| u_1(x) & u_2(x) \\
| |
| u_1'(x) & u_2'(x) \end{pmatrix}
| |
| \begin{pmatrix}
| |
| A'(x) \\
| |
| B'(x)\end{pmatrix} =
| |
| \begin{pmatrix}
| |
| 0\\
| |
| f\end{pmatrix}</math>
| |
| we can solve for (''A''′(''x''), ''B''′(''x''))<sup>''T''</sup>, so
| |
| :<math>\begin{pmatrix}
| |
| A'(x) \\
| |
| B'(x)\end{pmatrix}=
| |
| \begin{pmatrix}
| |
| u_1(x) & u_2(x) \\
| |
| u_1'(x) & u_2'(x) \end{pmatrix}^{-1}
| |
| \begin{pmatrix}
| |
| 0\\
| |
| f\end{pmatrix}</math>
| |
| :<math>={1\over W}
| |
| \begin{pmatrix}
| |
| u_2'(x) & -u_2(x) \\
| |
| -u_1'(x) & u_1(x) \end{pmatrix}
| |
| \begin{pmatrix}
| |
| 0\\
| |
| f\end{pmatrix},</math>
| |
| 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>A'(x) = - {1\over W} u_2(x) f(x),\; B'(x) = {1 \over W} u_1(x)f(x)</math>
| |
| :<math>A(x) = - \int {1\over W} u_2(x) f(x)\,dx,\; B(x) = \int {1 \over W} u_1(x)f(x)\,dx.</math>
| |
| | |
| 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 <math>A(x)</math> and <math> B(x)</math> are each determined only up to an arbitrary additive constant (the [[constant of integration]]); one would expect two constants of integration because the original equation was second order. 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 <math>L</math> is [[linear]].
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| * {{cite book | last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations | publisher=[[McGraw-Hill]] | location=New York | year=1955}}
| |
| * {{cite book | title = Elementary Differential Equations and Boundary Value Problems 8th Edition | first1 = W. E. | last1 = Boyce | first2 = R. C. | last2 = DiPrima | publisher = Wiley Interscience | year = 1965}}, pages 186-192, 237-241
| |
| * {{cite book
| |
| | surname = Teschl
| |
| | given = Gerald
| |
| |authorlink=Gerald Teschl
| |
| | title = Ordinary Differential Equations and Dynamical Systems
| |
| | publisher=[[American Mathematical Society]]
| |
| | place = [[Providence, Rhode Island|Providence]]
| |
| | year =
| |
| | url = http://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]].
| |
| *[http://planetmath.org/encyclopedia/VariationOfParameters.html PlanetMath page].
| |
| *[http://www.math-cs.ucmo.edu/~mjms/2007.1/srinivasan10.pdf Motivation of method via celestial mechanics]
| |
| | |
| [[Category:Ordinary differential equations]]
| |