Method of variation of parameters

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In mathematics, variation of parameters is a technique used to solve inhomogeneous ordinary differential equations. In our examples we will use second order equations for convenience, but the method works for equations of any order, as well as systems of equations.

Technique

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.

Suppose we have two linearly independent solutions to the given differential equation, u₁ and u₂. Let W be the Wronskian of these two functions, and W must be nonzero since we have supposed the solutions are linearly independent.

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
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x) \end\beginA'(x) \\B'(x)\end =\begin0\\f\end.]
Expanding,
[\beginA'(x)u_1(x)+B'(x)u_2(x)\\A'(x)u_1'(x)+B'(x)u_2'(x)\end =\begin0\\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\beginA'(x) \\B'(x)\end =\begin0\\f\end]
we can solve for (A′(x), B′(x))^T, so
[\beginA'(x) \\B'(x)\end=\begin
u_1(x) & u_2(x) \\
u_1'(x) & u_2'(x) \end^\begin0\\f\end]
[=\begin
u_2'(x) & -u_2(x) \\-u_1'(x) & u_1(x) \end\begin0\\f\end.]

So,
[A'(x) = - u_2(x) f(x),\; B'(x) = u_1(x)f(x)]
[A(x) = - \int u_2(x) f(x)\,dx,\; B(x) = \int 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 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.

Example usage
Let us solve
[ y''+4y'+4y=\cosh.\;\!]

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.\;\!]
Form the characteristic equation
[\lambda^2+4\lambda+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 u₁=e^-2x, and u₂=xe^{-2x^{. The Wronskian of these two functions is
[\begin e^ & xe^ \\-2e^ & -e^(2x-1)\\\end = -e^e^(2x-1)+2xe^e^ ]
[= -e^(2x-1)+2xe^= (-2x+1+2x)e^ = e^.\;\!]

We seek functions A(x) and B(x) so A(x)u₁+B(x)u₂ is a general solution of the inhomogeneous equation. We need only calculate the integrals
[A(x) = - \int u_2(x) f(x)\,dx,\; B(x) = \int u_1(x)f(x)\,dx]
that is,
[A(x) = - \int } xe^ \cosh\,dx = - \int xe^\cosh\,dx = -e^x(9(x-1)+e^(3x-1))+C_1]
[B(x) = \int } e^ \cosh\,dx = \int e^\cosh\,dx =e^(3+e^)+C_2 ]
where [C_1] and [C_2] are constants of integration.

A note on first-order differential equations
Variation of parameters may also be used for first-order differential equations, though other methods such as integrating factors and undetermined coefficients are usually used to obtain the same results with considerably less effort.

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