Border color only on boxed equation

OK, per request. You most likely want to switch back to the normal color anyway, so

\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,xcolor}
\begin{document}
\begin{equation}
  \colorlet{oldcolor}{.}
  \color{magenta}
  \boxed{\color{oldcolor} F(x)=\int_{0}^{x}f(t)\,\mathrm{d}t}\color{oldcolor}
\end{equation}
\end{document}

Needless to say that there are packages like empheq, tcolorbox and in particular the combination of the two which take care of these things automatically.

I propose to define a colorboxed command, based on \fcolorbox, which has two mandatory argument – the colour of the frame and the content of the frame, and the background of the frame as an optional argument:

\documentclass[12pt]{article}
\usepackage{amsmath,amssymb}

\usepackage[svgnames]{xcolor}
\makeatletter
    \newcommand{\colorboxed}[3][white]{\fcolorbox{#2}{#1}{\m@th$\displaystyle#3$}}
\makeatother

\begin{document}

\begin{equation}
  \colorboxed{IndianRed}{F(x)=\int_{0}^{x}f(t)dt}
\end{equation}
\begin{equation}
  \colorboxed[LavenderBlush!30]{IndianRed}{F(x)=\int_{0}^{x}f(t)dt}
\end{equation}

\end{document}

For reference, here's a version that uses tcolorbox's tcbhighmath command:

\documentclass{article}
\usepackage[many]{tcolorbox}
\tcbset{
    every box/.style={
            highlight math style={sharp corners,colback=white,colframe=magenta},
        }
}
\begin{document}
\begin{equation}
    \tcbhighmath{F(x)=\int_{0}^{x}f(t)dt}
\end{equation}
\end{document}

You might like to see Section 17.1 of the tcolorbox documentation for further reading; in particular, you might like to see \tcboxmath.