- Method of variation of parameters
In

mathematics ,**variation of parameters**also known as**variation of constants**, is a general method to solve inhomogeneous linearordinary differential equation s. It was developed by the Italian-French mathematicianJoseph Louis Lagrange .For first-order inhomogeneous linear differential equations it's usually possible to find solutions via

integrating factor s or undetermined coefficients with considerably less effort, although those methods are rather heuristics that involve guessing and don't work for all inhomogenous linear differential equations.Given an ordinary non-homogeneous linear differential equation of order "n":$y^\{(n)\}(x)\; +\; sum\_\{i=0\}^\{n-1\}\; a\_i(x)\; y^\{(i)\}(x)\; =\; b(x)$. (i)let $y\_1(x),\; ldots,\; y\_n(x)$ be a fundamental system of the corresponding homogeneous equation:$y^\{(n)\}(x)\; +\; sum\_\{i=0\}^\{n-1\}\; a\_i(x)\; y^\{(i)\}(x)\; =\; 0.$ (ii)

Then a particular solution to the non-homogeneous equation is given by:$y\_p(x)\; =\; sum\_\{i=1\}^\{n\}\; c\_i(x)\; y\_i(x)$ (iii)where the $c\_i(x)$ are continuous functions which satisfy the equations:$sum\_\{i=1\}^\{n\}\; c\_i^\text{'}(x)\; y\_i^\{(j)\}(x)\; =\; 0\; ,\; mathrm\{,\}\; quad\; j\; =\; 0,ldots,\; n-2$ (iv)

(results from substitution of (iii) into the homogeneous case (ii); )

and:$sum\_\{i=1\}^\{n\}\; c\_i^\text{'}(x)\; y\_i^\{(n-1)\}(x)\; =\; b(x).$. (v)

(results from substitution of (iii) into (i) and applying (iv);

$c\_i\text{'}(x)=0$ for all x and i is the only way to satisfy the condition, since all $y\_i(x)$ are linearly independent. It implies that all $c\_i(x)$ are independent of x in the homogeneous case b(x)=0. )

This linear system of "n" equations can then be solved using

Cramer's rule yielding:$c\_i^\text{'}(x)\; =\; frac\{W\_i(x)\}\{W(x)\}\; ,\; mathrm\{,\}\; quad\; i=1,ldots,n$where $W(x)$ is theWronskian determinant of the fundamental system and $W\_i(x)$ is the Wronskian determinant of the fundamental system with the "i"-th column replaced by $(0,\; 0,\; ldots,\; b(x)).$The particular solution to the non-homogeneous equation can then be written as:$sum\_\{i=1\}^n\; int\; frac\{W\_i(x)\}\{W(x)\}\; dx\; ,\; y\_i(x)$.

**Examples****Specific second order equation**Let us solve : $y"+4y\text{\'}+4y=cosh\{x\}.;!$

We want to find the general solution to the differential equation, that is, we want to find solutions to the homogeneous differential equation: $y"+4y\text{\'}+4y=0.;!$Form the characteristic equation: $lambda^2+4lambda+4=(lambda+2)^2=0;!$: $lambda=-2,-2.;!$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"

_{1}="e"^{-2"x"}, and "u"_{2}="xe"^{-2"x"}. TheWronskian of these two functions is : $egin\{vmatrix\}\; e^\{-2x\}\; xe^\{-2x\}\; \backslash -2e^\{-2x\}\; -e^\{-2x\}(2x-1)\backslash end\{vmatrix\}\; =\; -e^\{-2x\}e^\{-2x\}(2x-1)+2xe^\{-2x\}e^\{-2x\}$:$=\; -e^\{-4x\}(2x-1)+2xe^\{-4x\}=\; (-2x+1+2x)e^\{-4x\}\; =\; e^\{-4x\}.;!$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"

_{1}+"B"("x")"u"_{2}is a general solution of the inhomogeneous equation. We need only calculate the integrals:$A(x)\; =\; -\; int\; \{1over\; W\}\; u\_2(x)\; f(x),dx,;\; B(x)\; =\; int\; \{1\; over\; W\}\; u\_1(x)f(x),dx$that is,:$A(x)\; =\; -\; int\; \{1over\; e^\{-4x\; xe^\{-2x\}\; cosh\{x\},dx\; =\; -\; int\; xe^\{2x\}cosh\{x\},dx\; =\; -\{1over\; 18\}e^x(9(x-1)+e^\{2x\}(3x-1))+C\_1$:$B(x)\; =\; int\; \{1\; over\; e^\{-4x\; e^\{-2x\}\; cosh\{x\},dx\; =\; int\; e^\{2x\}cosh\{x\},dx\; =\{1over\; 6\}e^\{x\}(3+e^\{2x\})+C\_2$where $C\_1$ and $C\_2$ are constants of integration.**General second order equation**We have a differential equation of the form:$u"+p(x)u\text{\'}+q(x)u=f(x),$and we define the linear operator :$L=D^2+p(x)D+q(x),$where "D" represents the

differential operator . We therefore have to solve the equation $L\; u(x)=f(x)$ for $u(x)$, where $L$ and $f(x)$ are known.We must solve first the corresponding homogeneous equation::$u"+p(x)u\text{\'}+q(x)u=0,$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"

_{1}and "u"_{2}— we can proceed with variation of parameters.Now, we seek the general solution to the differential equation $u\_G(x)$ which we assume to be of the form:$u\_G(x)=A(x)u\_1(x)+B(x)u\_2(x).,$

Here, $A(x)$ and $B(x)$ are unknown and $u\_1(x)$ and $u\_2(x)$ are the solutions to the homogeneous equation. Observe that if $A(x)$ and $B(x)$ are constants, then $Lu\_G(x)=0$. We desire "A"="A"("x") and "B"="B"("x") to be of the form :$A\text{'}(x)u\_1(x)+B\text{'}(x)u\_2(x)=0.,$

Now,:$u\_G\text{'}(x)=(A(x)u\_1(x)+B(x)u\_2(x))\text{'}=(A(x)u\_1(x))\text{'}+(B(x)u\_2(x))\text{'},$:$=A\text{'}(x)u\_1(x)+A(x)u\_1\text{'}(x)+B\text{'}(x)u\_2(x)+B(x)u\_2\text{'}(x),$:$=A\text{'}(x)u\_1(x)+B\text{'}(x)u\_2(x)+A(x)u\_1\text{'}(x)+B(x)u\_2\text{'}(x),$and since we have required the above condition, then we have :$u\_G\text{'}(x)=A(x)u\_1\text{'}(x)+B(x)u\_2\text{'}(x).,$Differentiating again (omitting intermediary steps):$u\_G"(x)=A(x)u\_1"(x)+B(x)u\_2"(x)+A\text{\'}(x)u\_1\text{\'}(x)+B\text{\'}(x)u\_2\text{\'}(x).,$

Now we can write the action of "L" upon "u"

_{"G"}as:$Lu\_G=A(x)Lu\_1(x)+B(x)Lu\_2(x)+A\text{'}(x)u\_1\text{'}(x)+B\text{'}(x)u\_2\text{'}(x).,$Since "u"_{1}and "u"_{2}are solutions, then:$Lu\_G=A\text{'}(x)u\_1\text{'}(x)+B\text{'}(x)u\_2\text{'}(x).,$We have the system of equations:$egin\{pmatrix\}u\_1(x)\; u\_2(x)\; \backslash u\_1\text{'}(x)\; u\_2\text{'}(x)\; end\{pmatrix\}egin\{pmatrix\}A\text{'}(x)\; \backslash B\text{'}(x)end\{pmatrix\}\; =egin\{pmatrix\}0\backslash fend\{pmatrix\}.$Expanding, :$egin\{pmatrix\}A\text{'}(x)u\_1(x)+B\text{'}(x)u\_2(x)\backslash A\text{'}(x)u\_1\text{'}(x)+B\text{'}(x)u\_2\text{'}(x)end\{pmatrix\}\; =egin\{pmatrix\}0\backslash fend\{pmatrix\}.$So the above system determines precisely the conditions:$A\text{'}(x)u\_1(x)+B\text{'}(x)u\_2(x)=0,$:$A\text{'}(x)u\_1\text{'}(x)+B\text{'}(x)u\_2\text{'}(x)=Lu\_G=f.,$

We seek "A"("x") and "B"("x") from these conditions, so, given:$egin\{pmatrix\}u\_1(x)\; u\_2(x)\; \backslash u\_1\text{'}(x)\; u\_2\text{'}(x)\; end\{pmatrix\}egin\{pmatrix\}A\text{'}(x)\; \backslash B\text{'}(x)end\{pmatrix\}\; =egin\{pmatrix\}0\backslash fend\{pmatrix\}$we can solve for ("A"′("x"), "B"′("x"))

^{"T"}, so:$egin\{pmatrix\}A\text{'}(x)\; \backslash B\text{'}(x)end\{pmatrix\}=egin\{pmatrix\}u\_1(x)\; u\_2(x)\; \backslash u\_1\text{'}(x)\; u\_2\text{'}(x)\; end\{pmatrix\}^\{-1\}egin\{pmatrix\}0\backslash fend\{pmatrix\}$:$=\{1over\; W\}egin\{pmatrix\}u\_2\text{'}(x)\; -u\_2(x)\; \backslash -u\_1\text{'}(x)\; u\_1(x)\; end\{pmatrix\}egin\{pmatrix\}0\backslash fend\{pmatrix\},$where "W" denotes theWronskian 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,:$A\text{'}(x)\; =\; -\; \{1over\; W\}\; u\_2(x)\; f(x),;\; B\text{'}(x)\; =\; \{1\; over\; W\}\; u\_1(x)f(x)$:$A(x)\; =\; -\; int\; \{1over\; W\}\; u\_2(x)\; f(x),dx,;\; B(x)\; =\; int\; \{1\; over\; W\}\; u\_1(x)f(x),dx.$

Whilst 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 $A(x)$ and $B(x)$ 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 $A(x)$ or $B(x)$ does not change the value of $Lu\_G(x)$ because $L$ islinear .=Literature="Elementary Differntial Equations and Boundary Value problems ", W.E. Boyce and R.C. DiPrima, Wiley Interscience, 1965

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