Variation Of Parameters - Revision history

Khanh at 01:53, 6 November 2021

2021-11-06T01:53:20Z

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Licensing

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Higher Order

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Higher Order

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Higher Order

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Description of method

Lila: /* The Method of Variation of Parameters */

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The Method of Variation of Parameters

Lila at 21:21, 26 October 2021

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Description of method

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History

← Older revision		Revision as of 01:53, 6 November 2021
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	<math>\begin{align} \quad y(t) = \left ( \ln \mid \sec t \mid + C \right ) (1) + \left ( \cos t + D \right ) \cos t + \left ( \sin t + \ln \mid \sec t + \tan t \mid + E \right ) \sin t \end{align}</math>		<math>\begin{align} \quad y(t) = \left ( \ln \mid \sec t \mid + C \right ) (1) + \left ( \cos t + D \right ) \cos t + \left ( \sin t + \ln \mid \sec t + \tan t \mid + E \right ) \sin t \end{align}</math>
−
−	~~==References==~~
−	~~#Coddington, Earl A.; Levinson, Norman (1955). Theory of Ordinary Differential Equations. McGraw-Hill.~~
−	~~#Boyce, William E.; DiPrima, Richard C. (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). Wiley. pp. 186–192, 237–241.~~
−	~~#Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. American Mathematical Society.~~

	== Licensing ==		== Licensing ==

← Older revision		Revision as of 21:49, 26 October 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [https://en.wikipedia.org/wiki/Variation_of_parameters Variation of parameters, Wikipedia] under a CC BY-SA license		* [https://en.wikipedia.org/wiki/Variation_of_parameters Variation of parameters, Wikipedia] under a CC BY-SA license
		+	* [https://en.wikipedia.org/wiki/Variation_of_parameters The Method of Variation of Parameters, mathonline.wikidot.com] under a CC BY-SA license
		+	* [http://mathonline.wikidot.com/the-method-of-variation-of-parameters-for-higher-order-nonho Variation of Parameters for Higher Order, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 255: / Line 255: @@
 <math>\begin{align} \quad \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}}{dt^{n-1}} + ... + p_n(t) y = g(t) \end{align}</math>
-<p>We first solve the corresponding homogeneous differential equation to get $y_h(t) = C_1 y_1(t) + C_2y_2(t) + ... + C_ny_n(t)$. The functions $y_1(t)$, $y_2(t)$, &#8230;, $y_n(t)$ form a fundamental set. Assume a particular solution to the nonhomogeneous differential equation is of the form:</p>
+<p>We first solve the corresponding homogeneous differential equation to get  <math>y_h(t) = C_1 y_1(t) + C_2y_2(t) + ... + C_ny_n(t)  </math>. The functions  <math>y_1(t)  </math>,  <math>y_2(t)  </math>, &#8230;,  <math>y_n(t)  </math> form a fundamental set. Assume a particular solution to the nonhomogeneous differential equation is of the form:</p>
 <math>\begin{align} \quad Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t) + ... + u_n(t)y_n(t) \end{align}</math>
-<p>We then solve the following system of equations for the functions $u_1'(t)$, $u_2'(t)$, &#8230;, $u_n'(t)$.</p>
+<p>We then solve the following system of equations for the functions  <math>u_1'(t)  </math>,  <math>u_2'(t)  </math>, &#8230;,  <math>u_n'(t)  </math>.</p>
 <math>\begin{align} \quad \left\{\begin{matrix} u_1'(t)y_1(t) + u_2'(t)y_2(t) + ... + u_n'(t)y_n(t) = 0 \\ u_1'(t)y_1'(t) + u_2'(t)y_2'(t) + ... + u_n'(t)y_n'(t) = 0 \\ \vdots \\ u_1'(t)y_1^{(n-1)}(t) + u_2'(t)y_2^{(n-1)}(t) + ... + u_n'(t)y_n^{(n-1)}(t) = g(t) \end{matrix}\right. \end{align}</math>
-<p>Once again, a unique solution is guaranteed since $y_1(t)$, $y_2(t)$, &#8230;, $y_n(t)$ form a fundamental set of solutions implies the Wronskian $W(y_1, y_2, ..., y_n)$ is nonzero. Furthermore, it should be noted that the system above can be solved for with row reduction (if the process is simple) or more commonly by applying Cramer's rule once again.</p>
+<p>Once again, a unique solution is guaranteed since  <math>y_1(t)  </math>,  <math>y_2(t)  </math>, &#8230;,  <math>y_n(t)  </math> form a fundamental set of solutions implies the Wronskian  <math>W(y_1, y_2, ..., y_n)  </math> is nonzero. Furthermore, it should be noted that the system above can be solved for with row reduction (if the process is simple) or more commonly by applying Cramer's rule once again.</p>
-<p>We then integrate each of the functions $u_1'(t)$, $u_2'(t)$, &#8230;, $u_n'(t)$ to obtain $u_1(t)$, $u_2(t)$, &#8230;, $u_n(t)$. Lastly, we obtain that $Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t) + ... + u_n(t)y_n(t)$ as our particular solution.</p>
+<p>We then integrate each of the functions  <math>u_1'(t)  </math>,  <math>u_2'(t)  </math>, &#8230;,  <math>u_n'(t)  </math> to obtain  <math>u_1(t)  </math>,  <math>u_2(t)  </math>, &#8230;,  <math>u_n(t)  </math>. Lastly, we obtain that  <math>Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t) + ... + u_n(t)y_n(t)  </math> as our particular solution.</p>
 <p>We will now look at an example of using the method of variation of parameters for higher order nonhomogeneous differential equations.</p>
 ===Example 1===
-<p><strong>Solve the third order linear nonhomogeneous differential equation $y''' + y' = \tan t$ using the method of variation of parameters.</strong></p>
+<p><strong>Solve the third order linear nonhomogeneous differential equation  <math>y''' + y' = \tan t  </math> using the method of variation of parameters.</strong></p>
-<p>For the corresponding homogeneous differential equation $y''' + y' = 0$ we have that the characteristic equation is $r^3 + r = 0$ which can be factored as $r(r^2 + 1) = 0$ and so $r = 0$ or $r = \pm i$. The solution to the corresponding homogeneous differential equation is therefore:</p>
+<p>For the corresponding homogeneous differential equation  <math>y''' + y' = 0  </math> we have that the characteristic equation is  <math>r^3 + r = 0  </math> which can be factored as  <math>r(r^2 + 1) = 0  </math> and so  <math> r = 0 </math> or  <math>r = \pm i </math>. The solution to the corresponding homogeneous differential equation is therefore:</p>
 <math>\begin{align} \quad y(t) = P + Q\cos t + R \sin t \end{align}</math>
-<p>Furthermore, $y_1 = e^{0t} = 1$, $y_2 = e^{0t} \cos t= \cos t$, and $y_3 = e^{0t} \sin t = \sin t$.</p>
+<p>Furthermore,  <math>y_1 = e^{0t} = 1  </math>,  <math>y_2 = e^{0t} \cos t= \cos t  </math>, and  <math>y_3 = e^{0t} \sin t = \sin t  </math>.</p>
 <p>Now let's try to find a particular solution to this differential equation. We have that:</p>
@@ Line 276: / Line 276: @@
 <math>\begin{align} \quad u_1' (1) + u_2' \cos t + u_3' \sin t = 0 \\ \quad u_1' (0) + u_2' (-\sin t) + u_3' (\cos t) = 0 \\ \quad u_1' (0) + u_2' (\cos t) + u_3' (\sin t) = \tan t \end{align}</math>
-<p>We will use Cramer's rule in order to solve for $u_1'$, $u_2'$, and $u_3'$. We first find the corresponding Wronskian:</p>
+<p>We will use Cramer's rule in order to solve for  <math>u_1'  </math>,  <math>u_2'  </math>, and  <math>u_3'  </math>. We first find the corresponding Wronskian:</p>
 <math>\begin{align} \quad W = \begin{bmatrix} 1 & \cos t & \sin t \\ 0 & -\sin t & \cos t \\ 0 & -\cos t & -\sin t \end{bmatrix} = \begin{bmatrix} -\sin t & \cos t \\ -\cos t & -\sin t \end{bmatrix} = 1 \end{align}</math>
@@ Line 286: / Line 286: @@
 <math>\begin{align} \quad u_3' = \frac{\begin{bmatrix} 1 & \cos t & 0 \\ 0 & -\sin t & 0 \\ 0 & -\cos t & \tan t \end{bmatrix}}{1} = \begin{bmatrix} -\sin t & 0 \\ -\cos t & \tan t \end{bmatrix} = -\sin t \tan t = -\frac{\sin^2 t}{\cos t} \end{align}</math>
-<p>We will now integrate $u_1'$, $u_2'$, and $u_3'$ to get:</p>
+<p>We will now integrate  <math>u_1' </math>,  <math>u_2'  </math>, and  <math>u_3' </math> to get:</p>
 <math>\begin{align} \quad \int u_1'(t)   dt = \int \tan t   dt = \ln \mid \sec t \mid + C \end{align}</math>

@@ Line 288: / Line 288: @@
 <p>We will now integrate $u_1'$, $u_2'$, and $u_3'$ to get:</p>
-<math>\begin{align} \quad \int u_1'(t) \: dt = \int \tan t \: dt = \ln \mid \sec t \mid + C \end{align}</math>
+<math>\begin{align} \quad \int u_1'(t)   dt = \int \tan t   dt = \ln \mid \sec t \mid + C \end{align}</math>
-<math>\begin{align} \quad \int u_2'(t) \: dt = \int - \sin t \: dt = \cos t + D \end{align}</math>
+<math>\begin{align} \quad \int u_2'(t)   dt = \int - \sin t   dt = \cos t + D \end{align}</math>
-<math>\begin{align} \quad \int u_3'(t) \: dt = \int -\frac{\sin^2 t}{\cos t} \: dt = \int \frac{\cos^2 t - 1}{\cos t} \: dt = \int \left ( \cos t - \sec t \right ) \: dt = \sin t + \ln \mid \sec t + \tan t \mid + E \end{align}</math>
+<math>\begin{align} \quad \int u_3'(t)   dt = \int -\frac{\sin^2 t}{\cos t}   dt = \int \frac{\cos^2 t - 1}{\cos t}   dt = \int \left ( \cos t - \sec t \right )   dt = \sin t + \ln \mid \sec t + \tan t \mid + E \end{align}</math>
 <p>Thus we have that:</p>

← Older revision		Revision as of 21:35, 26 October 2021
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	==Higher Order==		==Higher Order==
		+	<p>The method of variation of parameters can also be applied to higher order differential equations. The process can be derived similarly. Suppose that we have a higher order differential equation of the following form:</p>

		+	<math>\begin{align} \quad \frac{d^ny}{dt^n} + p_1(t) \frac{d^{n-1}}{dt^{n-1}} + ... + p_n(t) y = g(t) \end{align}</math>
		+	<p>We first solve the corresponding homogeneous differential equation to get $y_h(t) = C_1 y_1(t) + C_2y_2(t) + ... + C_ny_n(t)$. The functions $y_1(t)$, $y_2(t)$, …, $y_n(t)$ form a fundamental set. Assume a particular solution to the nonhomogeneous differential equation is of the form:</p>
		+
		+	<math>\begin{align} \quad Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t) + ... + u_n(t)y_n(t) \end{align}</math>
		+	<p>We then solve the following system of equations for the functions $u_1'(t)$, $u_2'(t)$, …, $u_n'(t)$.</p>
		+
		+	<math>\begin{align} \quad \left\{\begin{matrix} u_1'(t)y_1(t) + u_2'(t)y_2(t) + ... + u_n'(t)y_n(t) = 0 \\ u_1'(t)y_1'(t) + u_2'(t)y_2'(t) + ... + u_n'(t)y_n'(t) = 0 \\ \vdots \\ u_1'(t)y_1^{(n-1)}(t) + u_2'(t)y_2^{(n-1)}(t) + ... + u_n'(t)y_n^{(n-1)}(t) = g(t) \end{matrix}\right. \end{align}</math>
		+	<p>Once again, a unique solution is guaranteed since $y_1(t)$, $y_2(t)$, …, $y_n(t)$ form a fundamental set of solutions implies the Wronskian $W(y_1, y_2, ..., y_n)$ is nonzero. Furthermore, it should be noted that the system above can be solved for with row reduction (if the process is simple) or more commonly by applying Cramer's rule once again.</p>
		+	<p>We then integrate each of the functions $u_1'(t)$, $u_2'(t)$, …, $u_n'(t)$ to obtain $u_1(t)$, $u_2(t)$, …, $u_n(t)$. Lastly, we obtain that $Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t) + ... + u_n(t)y_n(t)$ as our particular solution.</p>
		+	<p>We will now look at an example of using the method of variation of parameters for higher order nonhomogeneous differential equations.</p>
		+	===Example 1===
		+	<p><strong>Solve the third order linear nonhomogeneous differential equation $y''' + y' = \tan t$ using the method of variation of parameters.</strong></p>
		+	<p>For the corresponding homogeneous differential equation $y''' + y' = 0$ we have that the characteristic equation is $r^3 + r = 0$ which can be factored as $r(r^2 + 1) = 0$ and so $r = 0$ or $r = \pm i$. The solution to the corresponding homogeneous differential equation is therefore:</p>
		+
		+	<math>\begin{align} \quad y(t) = P + Q\cos t + R \sin t \end{align}</math>
		+	<p>Furthermore, $y_1 = e^{0t} = 1$, $y_2 = e^{0t} \cos t= \cos t$, and $y_3 = e^{0t} \sin t = \sin t$.</p>
		+	<p>Now let's try to find a particular solution to this differential equation. We have that:</p>
		+
		+	<math>\begin{align} \quad y_p = u_1y_1 + u_2y_2 + u_3y_3 \\ \quad y_p = u_1 + u_2 \cos t + u_3 \sin t \end{align}</math>
		+	<p>Thus we want to solve the following system of equations:</p>
		+
		+	<math>\begin{align} \quad u_1' (1) + u_2' \cos t + u_3' \sin t = 0 \\ \quad u_1' (0) + u_2' (-\sin t) + u_3' (\cos t) = 0 \\ \quad u_1' (0) + u_2' (\cos t) + u_3' (\sin t) = \tan t \end{align}</math>
		+	<p>We will use Cramer's rule in order to solve for $u_1'$, $u_2'$, and $u_3'$. We first find the corresponding Wronskian:</p>
		+
		+	<math>\begin{align} \quad W = \begin{bmatrix} 1 & \cos t & \sin t \\ 0 & -\sin t & \cos t \\ 0 & -\cos t & -\sin t \end{bmatrix} = \begin{bmatrix} -\sin t & \cos t \\ -\cos t & -\sin t \end{bmatrix} = 1 \end{align}</math>
		+	<p>Now we have that:</p>
		+
		+	<math>\begin{align} \quad u_1' = \frac{\begin{bmatrix} 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \\ \tan t & -\cos t & -\sin t \end{bmatrix}}{1} = \tan t \begin{bmatrix} \cos t & \sin t \\ -\sin t & \cos t \end{bmatrix} = \tan t \end{align}</math>
		+
		+	<math>\begin{align} \quad u_2' = \frac{\begin{bmatrix} 1 & 0 & \sin t \\ 0 & 0 & \cos t \\ 0 & \tan t & -\sin t \end{bmatrix}}{1} = - \begin{bmatrix} 1 & 0 & \sin t \\ 0 & \tan t & -\sin t \\ 0 & 0 & \cos t \end{bmatrix} = -\tan t \cos t = -\frac{\sin t}{\cos t} \cos t = -\sin t \end{align}</math>
		+
		+	<math>\begin{align} \quad u_3' = \frac{\begin{bmatrix} 1 & \cos t & 0 \\ 0 & -\sin t & 0 \\ 0 & -\cos t & \tan t \end{bmatrix}}{1} = \begin{bmatrix} -\sin t & 0 \\ -\cos t & \tan t \end{bmatrix} = -\sin t \tan t = -\frac{\sin^2 t}{\cos t} \end{align}</math>
		+	<p>We will now integrate $u_1'$, $u_2'$, and $u_3'$ to get:</p>
		+
		+	<math>\begin{align} \quad \int u_1'(t) \: dt = \int \tan t \: dt = \ln \mid \sec t \mid + C \end{align}</math>
		+
		+	<math>\begin{align} \quad \int u_2'(t) \: dt = \int - \sin t \: dt = \cos t + D \end{align}</math>
		+
		+	<math>\begin{align} \quad \int u_3'(t) \: dt = \int -\frac{\sin^2 t}{\cos t} \: dt = \int \frac{\cos^2 t - 1}{\cos t} \: dt = \int \left ( \cos t - \sec t \right ) \: dt = \sin t + \ln \mid \sec t + \tan t \mid + E \end{align}</math>
		+	<p>Thus we have that:</p>
		+
		+	<math>\begin{align} \quad y(t) = \left ( \ln \mid \sec t \mid + C \right ) (1) + \left ( \cos t + D \right ) \cos t + \left ( \sin t + \ln \mid \sec t + \tan t \mid + E \right ) \sin t \end{align}</math>

	==References==		==References==

@@ Line 33: / Line 33: @@
 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 ==
+== Description of method (Second Order)==
 <p>Consider a general second order linear nonhomogeneous differential equation whose coefficient functions <math>p</math>, <math>q</math>, and <math>g</math> are continuous:</p>

← Older revision		Revision as of 21:23, 26 October 2021
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	== Description of method ==		== Description of method ==

−	~~==The Method of Variation of Parameters==~~
	<p>Consider a general second order linear nonhomogeneous differential equation whose coefficient functions <math>p</math>, <math>q</math>, and <math>g</math> are continuous:</p>		<p>Consider a general second order linear nonhomogeneous differential equation whose coefficient functions <math>p</math>, <math>q</math>, and <math>g</math> are continuous:</p>

@@ Line 63: / Line 63: @@
 <p>We now integrate both sides of each equation above, and for constants <math>A</math> and <math>B</math>, we get <math>u_1(t)</math> and <math>u_2(t)</math>:</p>
-<math>\begin{align} \quad u_1(t) = -\int \frac{y_2(t)g(t)}{W(y_1, y_2)} \: dt + A \quad , \quad u_2(t) = \int \frac{y_1(t)g(t)}{W(y_1, y_2)} \: dt + B \end{align}</math>
+<math>\begin{align} \quad u_1(t) = -\int \frac{y_2(t)g(t)}{W(y_1, y_2)}  dt + A \quad , \quad u_2(t) = \int \frac{y_1(t)g(t)}{W(y_1, y_2)}  dt + B \end{align}</math>
 <p>Therefore, a particular solution to our second order linear nonhomogeneous differential equation is:</p>
-<math>\begin{align} \quad Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t) \\ \quad Y(t) = y_1(t) \left ( -\int \frac{y_2(t)g(t)}{W(y_1, y_2)} \: dt + A \right ) + y_2(t) \left ( \int \frac{y_1(t)g(t)}{W(y_1, y_2)} \: dt + B \right ) \\ \quad Y(t) = -y_1(t) \int \frac{y_2(t)g(t)}{W(y_1, y_2)} \: dt + y_2(t) \int \frac{y_1(t)g(t)}{W(y_1, y_2)} \: dt \: \underbrace{-Ay_1(t) + By_2(t)}_{=0} \\ \quad Y(t) = -y_1(t) \int \frac{y_2(t)g(t)}{W(y_1, y_2)} \: dt + y_2(t) \int \frac{y_1(t)g(t)}{W(y_1, y_2)} \: dt \end{align}</math>
+<math>\begin{align} \quad Y(t) = u_1(t)y_1(t) + u_2(t)y_2(t) \\ \quad Y(t) = y_1(t) \left ( -\int \frac{y_2(t)g(t)}{W(y_1, y_2)}  dt + A \right ) + y_2(t) \left ( \int \frac{y_1(t)g(t)}{W(y_1, y_2)}  dt + B \right ) \\ \quad Y(t) = -y_1(t) \int \frac{y_2(t)g(t)}{W(y_1, y_2)}  dt + y_2(t) \int \frac{y_1(t)g(t)}{W(y_1, y_2)}  dt  \underbrace{-Ay_1(t) + By_2(t)}_{=0} \\ \quad Y(t) = -y_1(t) \int \frac{y_2(t)g(t)}{W(y_1, y_2)}  dt + y_2(t) \int \frac{y_1(t)g(t)}{W(y_1, y_2)}  dt \end{align}</math>
 <p>And finally, the general solution to our differential equation will be:</p>
-<math>\begin{align} \quad y = Cy_1(t) + Dy_2(t) + Y(t) \\ \quad y = Cy_1(t) + Dy_2(t) -y_1(t) \int \frac{y_2(t)g(t)}{W(y_1, y_2)} \: dt + y_2(t) \int \frac{y_1(t)g(t)}{W(y_1, y_2)} \: dt \end{align}</math>
+<math>\begin{align} \quad y = Cy_1(t) + Dy_2(t) + Y(t) \\ \quad y = Cy_1(t) + Dy_2(t) -y_1(t) \int \frac{y_2(t)g(t)}{W(y_1, y_2)}  dt + y_2(t) \int \frac{y_1(t)g(t)}{W(y_1, y_2)}  dt \end{align}</math>
 <p>Note that the method of variation of parameters is useful provided that the general solution to the corresponding second order linear homogeneous differential equation is easy to solve, and provided that the two integrals in the formula above are relatively simply to compute.</p>

← Older revision		Revision as of 21:10, 26 October 2021
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	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.		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 first sketched by the Swiss mathematician Leonhard Euler (1707–1783), and later completed by the Italian-French mathematician Joseph-Louis Lagrange (1736–1813).
−
−	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. In his 1749 study of the motions of the earth, Euler obtained differential equations for the orbital elements. In 1753, he applied the method to his study of the motions of the moon.
−
−	Lagrange first used the method in 1766. Between 1778 and 1783, he further developed the method in two series of memoirs: one on variations in the motions of the planets and another on determining the orbit of a comet from three observations. During 1808–1810, Lagrange gave the method of variation of parameters its final form in a third series of papers.

	== Intuitive explanation ==		== Intuitive explanation ==