In mathematics, variation of parameters, also known as variation of constants, is a general method to solve

inhomogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, si ...

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...

s. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via

integrating factor In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calc ...

s or undetermined coefficients with considerably less effort, although those methods leverage

heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediat ...

s that involve guessing and do not work for all inhomogeneous linear differential equations. Variation of parameters extends to linear

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation,

wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...

, and

vibrating plate Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

equation. In this setting, the method is more often known as

Duhamel's principle In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equation, wave equation, and vibrating plate equation ...

, named after

Jean-Marie Duhamel Jean-Marie Constant Duhamel (; ; 5 February 1797 – 29 April 1872) was a French mathematician and physicist. His studies were affected by the troubles of the Napoleonic era. He went on to form his own school ''École Sainte-Barbe''. 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 Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

(1707–1783), and later completed by the Italian-French mathematician

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiapages 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
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
pages 809–816

Description of method

Given an ordinary non-homogeneous linear differential equation of order ''n'' Let

y_1(x), \ldots, y_n(x)

be a

fundamental system In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

of solutions of the corresponding homogeneous equation Then a

particular solution In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

to the non-homogeneous equation is given by where the

c_i(x)

are differentiable functions which are assumed to satisfy the conditions Starting with (), repeated differentiation combined with repeated use of () gives One last differentiation gives By substituting () into () and applying () and () it follows that The linear system ( and ) of ''n'' equations can then be solved using

Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants ...

yielding :

c_i'(x) = \frac, \, \quad i=1,\ldots,n

where

W(x)

is the

Wronskian determinant In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian o ...

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_^n y_i(x) \, \int \frac\, \mathrm dx.

Intuitive explanation

Consider the equation of the forced dispersionless spring, in suitable units: :

x''(t) + x(t) = F(t).

Here is the displacement of the spring from the equilibrium , and 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

t=s

and

t=s+ds

, the momentum corresponding to the solution has a net change

F(s)\,ds

(see:

Impulse (physics) In classical mechanics, impulse (symbolized by or Imp) is the integral of a force, , over the time interval, , for which it acts. Since force is a vector quantity, impulse is also a vector quantity. Impulse applied to an object produces an equ ...

). A solution to the inhomogeneous equation, at the present time , is obtained by linearly superposing the solutions obtained in this manner, for going between 0 and . The homogeneous initial-value problem, representing a small impulse

F(s)\,ds

being added to the solution at time

t=s

, is :

x''(t)+x(t)=0,\quad x(s)=0,\ x'(s)=F(s)\,ds.

The unique solution to this problem is easily seen to be

x(t) = F(s)\sin(t-s)\,ds

. The linear superposition of all of these solutions is given by the integral: :

x(t) = \int_0^t F(s)\sin(t-s)\,ds.

To verify that this satisfies the required equation: :

x'(t)=\int_0^t F(s)\cos(t-s)\,ds

x''(t) = F(t) - \int_0^tF(s)\sin(t-s)\,ds = F(t)-x(t),

as required (see:

Leibniz integral rule In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integral are

). The general method of variation of parameters allows for solving an inhomogeneous linear equation :

Lx(t)=F(t)

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

x_s

the solution of the homogeneous initial value problem :

Lx(t)=0, \quad x(s)=0,\ x'(s)=F (s)\,ds.

Then a particular solution of the inhomogeneous equation is :

x (t)=\int_0^t x_s (t)\,ds,

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

x_s

then being given in terms of explicit linear combinations of linearly independent fundamental solutions. In the case of the forced dispersionless spring, the kernel

\sin(t-s)=\sin t\cos s - \sin s\cos t

is the associated decomposition into fundamental solutions.

Examples

First-order equation

y' + p(x)y = q(x)

The complementary solution to our original (inhomogeneous) equation is the general solution of the corresponding homogeneous equation (written below): :

y' + p(x)y = 0

This homogeneous differential equation can be solved by different methods, for example

separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...

: :

\frac y + p(x)y = 0

\frac=-p(x)y

= -,

\int \frac \, dy = -\int p(x) \, dx

\ln , y,  = -\int p(x) \, dx + C

y = \pm e^ = C_0 e^

The complementary solution to our original equation is therefore: :

y_c = C_0 e^

Now we return to solving the non-homogeneous equation: :

y' + p(x)y = q(x)

Using the method variation of parameters, the particular solution is formed by multiplying the complementary solution by an unknown function ''C''(''x''): :

y_p = C(x) e^

By substituting the particular solution into the non-homogeneous equation, we can find ''C''(''x''): :

C' (x) e^ - C(x) p(x) e^ + p(x) C(x) e^ = q(x)

C' (x) e^ = q(x)

C' (x) = q(x) e^

C(x) =\int q(x) e^ \, dx + C_1

We only need a single particular solution, so we arbitrarily select

C_1=0

for simplicity. Therefore the particular solution is: :

y_p =e^ \int q(x) e^ \, dx

The final solution of the differential equation is: :

\begin
y &= y_c + y_p\\
&=C_0 e^ + e^ \int q(x) e^ \, dx
\end

This recreates the method of

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.

The characteristic equation is: :

\lambda^2+4\lambda+4=(\lambda+2)^2=0

Since

\lambda=-2

is a repeated root, we have to introduce a factor of ''x'' for one solution to ensure linear independence:

u_1 = e^

and

u_2 =x e^

. The

Wronskian In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian o ...

of these two functions is :

W=\begin
  e^ & xe^ \\
-2e^ & -e^(2x-1)\\
\end = -e^e^(2x-1)+2xe^e^ = e^.

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 particular solution of the non-homogeneous equation. We need only calculate the integrals :

A(x) = - \int  u_2(x) b(x)\,\mathrm dx,\; B(x) = \int  u_1(x)b(x)\,\mathrm dx

Recall that for this example :

b(x) = \cosh x

That is, :

A(x) = - \int  xe^ \cosh x \,\mathrm dx = - \int xe^\cosh x \,\mathrm dx = -e^x\left(9(x-1)+e^(3x-1)\right)+C_1

B(x) = \int  e^ \cosh x \,\mathrm dx = \int e^\cosh x\,\mathrm dx =e^x\left(3+e^\right)+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

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'+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

.) Since the above is only one equation and we have two unknown functions, it is reasonable to impose a second condition. We choose the following: :

A'(x)u_1(x)+B'(x)u_2(x)=0.

Now, :

\begin
u_G'(x) &= \left (A(x)u_1(x)+B(x)u_2(x) \right )' \\
&= \left (A(x)u_1(x) \right )'+ \left (B(x)u_2(x) \right )'\\
&=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) \\
&= A(x)u_1'(x)+B(x)u_2'(x)
\end

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 ''u''₁ and ''u''₂ are solutions, then :

Lu_G=A'(x)u_1'(x)+B'(x)u_2'(x).

We have the system of equations :

\begin
u_1(x)  & u_2(x) \\
u_1'(x) & u_2'(x) \end
\begin
A'(x) \\
B'(x)\end =
\begin 0 \\ f \end.

Expanding, :

\begin
A'(x)u_1(x)+B'(x)u_2(x)\\
A'(x)u_1'(x)+B'(x)u_2'(x) \end
= \begin 0\\f\end.

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
u_1(x)  & u_2(x) \\
u_1'(x) & u_2'(x)
\end
\begin
A'(x) \\
B'(x)\end =
\begin
0\\
f\end

we can solve for (''A''′(''x''), ''B''′(''x''))^T, so :

\begin A'(x) \\ B'(x) \end =
\begin
u_1(x)  & u_2(x) \\
u_1'(x) & u_2'(x)
\end^
\begin 0\\ f \end =\frac \begin
u_2'(x)  & -u_2(x) \\
-u_1'(x) & u_1(x) \end
\begin 0\\ f \end,

where ''W'' denotes the

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, :

\begin
A'(x) &= -  u_2(x) f(x), & B'(x) &=  u_1(x)f(x) \\
A(x)  &= - \int  u_2(x) f(x)\,\mathrm dx, & B(x) &= \int  u_1(x)f(x)\,\mathrm dx
\end

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

A(x)

and

B(x)

are each determined only up to an arbitrary additive constant (the

constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connect ...

). Adding a constant to

A(x)

B(x)

does not change the value of

Lu_G(x)

because the extra term is just a linear combination of ''u''₁ and ''u''₂, which is a solution of

L

by definition.

Notes

References

* * * {{cite book , last = Teschl , first = Gerald , author-link = Gerald Teschl , title = Ordinary Differential Equations and Dynamical Systems , publisher =

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

, year = 2012 , url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/

External links

Online Notes / Proof
by Paul Dawkins,

Lamar University Lamar University (Lamar or LU) is a public university in Beaumont, Texas. Lamar has been a member of the Texas State University System since 1995. It was the flagship institution of the former Lamar University System. As of the fall of 2021, t ...

.
PlanetMath page

A NOTE ON LAGRANGE’S METHOD OF VARIATION OF PARAMETERS
Ordinary differential equations