Variation of parameters

In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.

For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics 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 Leonard Euler (1707 - 1783) and 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; and 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, Lagrange further developed the method both in a series of memoirs on variations in the motions of the planets and in another series of memoirs on determining the orbit of a comet from three observations. (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.) During 1808-1810, Lagrange gave the method of variation of parameters its final form in a series of papers.

Description of method

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 solutions 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

$\sum _{i=1}^{n}c_{i}'(x)y_{i}^{(j)}(x)=0\,\mathrm {,} \quad j=0,\ldots ,n-2.$ (iv)

By substituting (iii) into (i) and applying (iv) it follows that

$\sum _{i=1}^{n}c_{i}'(x)y_{i}^{(n-1)}(x)=b(x).$ (v)

The linear system (v) of n equations can then be solved using Cramer's rule yielding

$c_{i}'(x)={\frac {W_{i}(x)}{W(x)}},\,\quad i=1,\ldots ,n$

where $W(x)$ is the Wronskian 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

$y''+4y'+4y=\cosh {x}.\;\!$

We want to find the general solution to the differential equation, that is, we want to find solutions to the homogeneous differential equation

$y''+4y'+4y=0.\;\!$

From the characteristic equation

$\lambda ^{2}+4\lambda +4=(\lambda +2)^{2}=0\;\!$

$\lambda =-2.\;\!$

Since we have a repeated root, we have to introduce a factor of x for one solution to ensure linear independence.

So, we obtain u₁ = e^−2x, and u₂ = xe^−2x. The Wronskian of these two functions is

${\begin{vmatrix}e^{-2x}&xe^{-2x}\\-2e^{-2x}&-e^{-2x}(2x-1)\\\end{vmatrix}}=-e^{-2x}e^{-2x}(2x-1)+2xe^{-2x}e^{-2x}$

$=-e^{-4x}(2x-1)+2xe^{-4x}=(-2x+1+2x)e^{-4x}=e^{-4x}.\;\!$

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)u₁ + B(x)u₂ is a general solution of the non-homogeneous equation. We need only calculate the integrals

$A(x)=-\int {1 \over W}u_{2}(x)b(x)\,dx,\;B(x)=\int {1 \over W}u_{1}(x)b(x)\,dx$

that is,

$A(x)=-\int {1 \over e^{-4x}}xe^{-2x}\cosh {x}\,dx=-\int xe^{2x}\cosh {x}\,dx=-{1 \over 18}e^{x}(9(x-1)+e^{2x}(3x-1))+C_{1}$

$B(x)=\int {1 \over e^{-4x}}e^{-2x}\cosh {x}\,dx=\int e^{2x}\cosh {x}\,dx={1 \over 6}e^{x}(3+e^{2x})+C_{2}$

where $C_{1}$ and $C_{2}$ are constants of integration.

General second order equation

We have a differential equation of the form

$u''+p(x)u'+q(x)u=f(x)\,$

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 $Lu(x)=f(x)$ for $u(x)$, where $L$ and $f(x)$ are known.

We must solve first the corresponding homogeneous equation:

$u''+p(x)u'+q(x)u=0\,$

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 u₁ and u₂ — 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

$A'(x)u_{1}(x)+B'(x)u_{2}(x)=0.\,$

Now,

$u_{G}'(x)=(A(x)u_{1}(x)+B(x)u_{2}(x))'=(A(x)u_{1}(x))'+(B(x)u_{2}(x))'\,$

$=A'(x)u_{1}(x)+A(x)u_{1}'(x)+B'(x)u_{2}(x)+B(x)u_{2}'(x)\,$

$=A'(x)u_{1}(x)+B'(x)u_{2}(x)+A(x)u_{1}'(x)+B(x)u_{2}'(x)\,$

and since we have required the above condition, then we have

$u_{G}'(x)=A(x)u_{1}'(x)+B(x)u_{2}'(x).\,$

Differentiating again (omitting intermediary steps)

$u_{G}''(x)=A(x)u_{1}''(x)+B(x)u_{2}''(x)+A'(x)u_{1}'(x)+B'(x)u_{2}'(x).\,$

Now we can write the action of L upon u_G as

$Lu_{G}=A(x)Lu_{1}(x)+B(x)Lu_{2}(x)+A'(x)u_{1}'(x)+B'(x)u_{2}'(x).\,$

Since u₁ and u₂ are solutions, then

$Lu_{G}=A'(x)u_{1}'(x)+B'(x)u_{2}'(x).\,$

We have the system of equations

${\begin{pmatrix}u_{1}(x)&u_{2}(x)\\u_{1}'(x)&u_{2}'(x)\end{pmatrix}}{\begin{pmatrix}A'(x)\\B'(x)\end{pmatrix}}={\begin{pmatrix}0\\f\end{pmatrix}}.$

Expanding,

${\begin{pmatrix}A'(x)u_{1}(x)+B'(x)u_{2}(x)\\A'(x)u_{1}'(x)+B'(x)u_{2}'(x)\end{pmatrix}}={\begin{pmatrix}0\\f\end{pmatrix}}.$

So the above system determines precisely the conditions

$A'(x)u_{1}(x)+B'(x)u_{2}(x)=0\,$

$A'(x)u_{1}'(x)+B'(x)u_{2}'(x)=Lu_{G}=f.\,$

We seek A(x) and B(x) from these conditions, so, given

${\begin{pmatrix}u_{1}(x)&u_{2}(x)\\u_{1}'(x)&u_{2}'(x)\end{pmatrix}}{\begin{pmatrix}A'(x)\\B'(x)\end{pmatrix}}={\begin{pmatrix}0\\f\end{pmatrix}}$

we can solve for (A′(x), B′(x))^T, so

${\begin{pmatrix}A'(x)\\B'(x)\end{pmatrix}}={\begin{pmatrix}u_{1}(x)&u_{2}(x)\\u_{1}'(x)&u_{2}'(x)\end{pmatrix}}^{-1}{\begin{pmatrix}0\\f\end{pmatrix}}$

$={1 \over W}{\begin{pmatrix}u_{2}'(x)&-u_{2}(x)\\-u_{1}'(x)&u_{1}(x)\end{pmatrix}}{\begin{pmatrix}0\\f\end{pmatrix}},$

where W denotes the Wronskian of u₁ and u₂. (We know that W is nonzero, from the assumption that u₁ and u₂ are linearly independent.)

So,

$A'(x)=-{1 \over W}u_{2}(x)f(x),\;B'(x)={1 \over W}u_{1}(x)f(x)$

$A(x)=-\int {1 \over W}u_{2}(x)f(x)\,dx,\;B(x)=\int {1 \over W}u_{1}(x)f(x)\,dx.$

While homogeneous equations are relatively easy to solve, this method allows the calculation of the coefficients of the general solution of the inhomogeneous 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 $L$ is linear.

References

• Coddington, Earl A.; Levinson, Norman (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill.
• Boyce, W. E.; DiPrima, R. C. (1965). Elementary Differential Equations and Boundary Value Problems 8th Edition. Wiley Interscience., pages 186-192, 237-241
• Teschl, Gerald. Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society.

External links

• Online Notes / Proof by Paul Dawkins, Lamar University.
• PlanetMath page.
• Motivation of method via celestial mechanics