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* 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 ==
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:<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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