variation of parameters
VariationOfParameters
2002-05-20 05:20:29
2005-03-06 20:19:41
Theorem
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The method of \emph{variation of parameters} is a way of finding a particular
solution to a nonhomogeneous linear differential equation.
Suppose that we have an $n$th order linear differential operator
\begin{equation*}
L[y] := y^{(n)} + p_1(t) y^{(n-1)} + \cdots + p_n(t) y ,
\end{equation*}
and a corresponding nonhomogeneous differential equation
\begin{equation}
\label{nonhom-diffeq}
L[y] = g(t).
\end{equation}
Suppose that we know a fundamental set of solutions
$y_1,y_2,\ldots,y_n$ of the corresponding homogeneous differential equation
$L[y_c]=0$. The general solution of the homogeneous equation is
\begin{equation*}
y_c(t) = c_1 y_1(t) + c_2 y_2(t) + \cdots + c_n y_n(t),
\end{equation*}
where $c_1,c_2,\ldots,c_n$ are constants.
The general solution to the nonhomogeneous equation $L[y]=g(t)$ is then
\begin{equation*}
y(t) = y_c(t) + Y(t),
\end{equation*}
where $Y(t)$ is a particular solution which satisfies $L[Y]=g(t)$, and the
constants $c_1,c_2,\ldots,c_n$ are chosen to satisfy the appropriate
boundary conditions or initial conditions.
The key step in using variation of parameters is to suppose that the
particular solution is given by
\begin{equation}
\label{part-sol-form}
Y(t) = u_1(t) y_1(t) + u_2(t) y_2(t) + \cdots + u_n(t) y_n(t),
\end{equation}
where $u_1(t),u_2(t),\ldots,u_n(t)$ are as yet to be determined functions
(hence the name \emph{variation of parameters}). To find
these $n$ functions we need a set of $n$ independent equations.
One obvious condition is that the proposed ansatz satisfies Eq.
\eqref{nonhom-diffeq}. Many possible additional conditions are possible,
we choose the ones that make further calculations easier. Consider the
following set of $n-1$ conditions
\begin{eqnarray*}
y_1 u_1' + y_2 u_2' + \cdots + y_n u_n' &=& 0 \\
y_1' u_1' + y_2' u_2' + \cdots + y_n' u_n' &=& 0 \\
&\vdots \\
y_1^{(n-2)} u_1' + y_2^{(n-2)} u_2' + \cdots + y_n^{(n-2)} u_n' &=& 0.
\end{eqnarray*}
Now, substituting Eq.~\eqref{part-sol-form} into $L[Y]=g(t)$ and using the
above conditions, we can get another equation
\begin{equation*}
y_1^{(n-1)} u_1' + y_2^{(n-1)} u_2' + \cdots + y_n^{(n-1)} u_n' = g .
\end{equation*}
So we have a system of $n$ equations for $u_1',u_2',\ldots,u_n'$ which
we can solve using Cramer's rule:
\begin{equation*}
u_m'(t) = \frac{g(t) W_m(t)}{W(t)}, \quad m=1,2,\ldots,n .
\end{equation*}
Such a solution always exists since the Wronskian $W=W(y_1,y_2,\ldots,y_n)$
of the system is nowhere zero, because the $y_1,y_2,\ldots,y_n$ form a
fundamental set of solutions. Lastly the term $W_m$ is the Wronskian
determinant with the $m$th column replaced by the column
$(0,0,\ldots,0,1)$.
Finally the particular solution can be written explicitly as
\begin{equation*}
Y(t) = \sum_{m=1}^n y_m(t) \int \frac{g(t) W_m(t)}{W(t)} dt .
\end{equation*}
\begin{thebibliography}{1}
\bibitem{Boyce}
W.~E.~Boyce and R.~C.~DiPrima.
\textit{Elementary Differential Equations and Boundary Value Problems}
John Wiley \& Sons, 6th edition, 1997.
\end{thebibliography}