Here is the compilable code:
\documentclass{article}
\usepackage{amsmath}
\usepackage[framemethod=TikZ]{mdframed}
%% the following is commaon for all examples in mdframed manual
\mdfsetup{skipabove=\topskip,skipbelow=\topskip}
\newrobustcmd\ExampleText{%
An \textit{inhomogeneous linear} differential equation has the form
\begin{align}
L[v ] = f,
\end{align}
where $L$ is a linear differential operator, $v$ is the dependent
variable, and $f$ is a given non-zero function of the independent
variables alone.
}
%%% upto here
\newcounter{theo}[section]
\newenvironment{theo}[1][]{%
\stepcounter{theo}%
\ifstrempty{#1}%
{\mdfsetup{%
frametitle={%
\tikz[baseline=(current bounding box.east),outer sep=0pt]
\node[anchor=east,rectangle,fill=blue!20]
{\strut Theorem~\thetheo};}}
}%
{\mdfsetup{%
frametitle={%
\tikz[baseline=(current bounding box.east),outer sep=0pt]
\node[anchor=east,rectangle,fill=blue!20]
{\strut Theorem~\thetheo:~#1};}}%
}%
\mdfsetup{innertopmargin=10pt,linecolor=blue!20,%
linewidth=2pt,topline=true,
frametitleaboveskip=\dimexpr-\ht\strutbox\relax,}
\begin{mdframed}[]\relax%
}{\end{mdframed}}
\begin{document}
\begin{theo}[Inhomogeneous Linear]
\ExampleText
\end{theo}
\begin{theo}
\ExampleText
\end{theo}
\end{document}

Compare it with the code you have assembled to see what is missing.
One thing which I forgot to mention and reminded by Karl, is the − in −\ht (frametitleaboveskip=\dimexpr−\ht\strutbox\relax to be precise) which is a unicode symbol (U+2212 char, thanks to Karl for the code) which should be minus sign.
\documentclass{...}and ending with\end{document}. – onewhaleid Feb 13 '15 at 06:21