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{\Large\bf Second Order Linear Ordinary Differential Equations}

\vspace{.3in}

A second order linear ODE has the form
   \[ S(x)y''(x) + P(x)y'(x) + Q(x)y(x) = F(x). \]
On any interval where \( S(x) = 0 \), the ODE has the form
\begin{equation}
  y''(x) + p(x)y'(x) + q(x)y(x) = f(x)      \label{sec:lin}
\end{equation}

Examples:
\begin{enumerate}
   \item A linear equation is
     \[ y''(x) + sin(x)y'(x) + x^2y(x) = cos(x) \]
   \item A nonlinear (but similar) equation is
     \[ y''(x) + sin(y)y'(x) + x^2y(x) = cos(y + x) \]
\end{enumerate}

Equation~\ref{sec:lin} is homogeneous if \( f(x) = 0 \) on the
interval of interest.  Otherwise, the equation is nonhomogeneous.
For example, \( y''(x) + sin(x)y'(x) + x^2y(x) = 0 \) is homogeneous
while the first example above is nonhomogeneous.

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