Align equal signs too much spacing in cases

Use aligned.

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation*}
\begin{cases}
\begin{aligned}
\dot{x} &= f(x) \\
x(0) &= x_0
\end{aligned}
\end{cases}
\end{equation*}

\end{document}

Now compare with

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation*}
\begin{cases}
\dot{x} = f(x) \\
x(0) = x_0
\end{cases}
\end{equation*}

\end{document}

I believe that the latter format is better.

Use the empheq package (needless to load amsmath in this case):

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{empheq}

\begin{document}

\begin{empheq}[left=\empheqlbrace]{align*}
\dot x & = f(x)\\%
x(0) & = x_0
\end{empheq}

\end{document} 

Here's a solution that doesn't require loading any packages.

\documentclass{article}
\begin{document}
\[
\left\{
\begin{array}{r @{{}={}} l}
  \dot{x} & f(x) \\
  x(0)    & x_0
\end{array}
\right.
\]  
\end{document}