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{\Large\bf Methods for Finding Particular Solutions}

\vspace{.3in}

Two basic procedures exists for finding a particular solution to
the second order nonhomogeneous problem
\begin{equation}
   y''(x) + p(x)y'(x) + q(x)y(x) = f(x).   \label{eq:nonhomo}
\end{equation}
The {\em method of undetermined coefficients} works well
when the equation has constant coefficients and the
nonhomogeneous term, f(x), is a finite linear combination of
finite products of exponential, trigonometric, and polynomial functions.
{\em Variation of parameters} always produces a particular
solution of equation (\ref{eq:nonhomo}) provided that the general
solution to the homogeneous problem is known.

\vspace{.6in}

\begin{itemize}
\item {\bf The method of undetermined coefficients} takes advantage
of the case when equation (\ref{eq:nonhomo}) has a special form.
First, the equation must have constant coefficients (i.e. p(x) and
q(x) are just constants).  Second, the forcing term, f(x), must
be a finite linear combination of finite products of
  \begin{itemize}
  \item  exponential functions $e^{rx}$,
  \item  sine and cosine functions $\cos(kx) \: \sin(mx)$,
  \item  and polynomial functions $a_{n}x^{n} + \cdots + a_{1}x + a_{0}$
  \end{itemize}
For example, 
f(x) = $(1 + x^{2})e^{2x} + (2x-9)e^{3x} \cos(x)\sin(2x)$.

\item {\bf Variation of parameters} will solve (your solution may
have to be expressed as an integral) any equation of form
(\ref{eq:nonhomo}) provided that the general solution of the
associated homogeneous problem can be found. 

\end{itemize}

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