Variation of parameters

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. It was developed by Joseph Louis Lagrange[citation needed].

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.

Contents

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 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 ci(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)

ci'(x) = 0 for all x and i is the only way to satisfy the condition, since all yi(x) are linearly independent. It implies that all ci(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'(x) = \frac{W_i(x)}{W(x)}, \, \quad i=1,\ldots,n

where W(x) is the Wronskian determinant of the fundamental system and Wi(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 C1 and C2 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 uG(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 u1(x) and u2(x) are the solutions to the homogeneous equation. Observe that if A(x) and B(x) are constants, then LuG(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 uG 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 LuG(x) because L is linear.

References

External links


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