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{\Large\bf Second Order Linear Nonhomogeneous Equations }

\vspace{.3in}

Now consider a second order equation of the form
\begin{equation}
   y''(x) + p(x)y'(x) + q(x)y(x) = f(x)   \label{eq:nonhomo}
\end{equation}
on some interval [a,b].

\vspace{.6in}

If $y_{c}(x)$ solves the homogeneous problem, % link here
and $y_{p}(x)$ solves equation (\ref{eq:nonhomo}), then letting
\mbox{$w(x) = y_{c}(x) + y_{p}(x)$} 
gives another solution of equation
\ref{eq:nonhomo} ( {\em Try this !} ).  This shows that the
general solution of the nonhomogeneous problem is 
\begin{equation}
  y(x) = y_{c}(x) + y_{p}(x)              \label{eq:non_gs}
\end{equation}
where $y_{c}(x)$ , called the {\em complimentary solution},
is the general solution of the homogeneous problem,
and $y_{p}(x)$, called
the {\em particular solution}, is any solution of the nonhomogeneous
equation (\ref{eq:nonhomo}).


\vspace{.4in}


The problem now is
\begin{itemize}
 \item finding the complimentary solution
 \item finding a particular solution
\end{itemize}


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