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\fbox{%
   \parbox{0.8\linewidth}{%
      \begin{align*}
         \rho'(\xi)&=-\frac 1 2 \quad ,\quad \rho(\xi_1)=\lambda_1(U_L)=v_L-\rho_L\\
         \rho(\xi)&=-\frac 1 2 \xi +c_1
      \end{align*}
   }
}

\Aboxed{\rho'(\xi)&=-\frac 1 2 \quad ,\quad \rho(\xi_1)=\lambda_1(U_L)=v_L-\rho_L}\\

\documentclass[12pt,a4paper,oneside,ngerman]{book}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{enumerate}
\usepackage{amstext}
\usepackage{empheq}
\usepackage{graphicx}
\usepackage{hhline}
\usepackage{float}
\usepackage{rotating}
\usepackage{subfig}
\begin{document}
\begin{align*}
\rho'(\xi)&=-\frac 1 2 \quad ,\quad \rho(\xi_1)=\lambda_1(U_L)=v_L-\rho_L\\
\rho(\xi)&=-\frac 1 2 \xi +c_1\\
\rho(\xi_1)&=\rho(v_L-\rho_L)=-\frac 1 2 (v_L-\rho_L)+c_1\stackrel{!}{=}\rho_L \quad \Rightarrow \quad c_1=\frac 1 2 (v_L+\rho_L)\\
\rho(\xi)&= -\frac 1 2 \xi + \frac 1 2 (v_L+\rho_L)\\\\
v'(\xi)&= \frac 1 2 \rho(\xi)= -\frac 1 4 \xi + \frac 1 4 (v_L+\rho_L) \quad,\quad v(\xi_1)=v_L\\
v(\xi)&= -\frac 1 8 \xi^2 + \frac 1 4 (v_L+\rho_L)\xi + c_2\\
v(\xi_1) &= -\frac 1 8 (v_L-\rho_L)^2 + \frac 1 4 (v_L+\rho_L)(v_L-\rho_L) + c_2\stackrel{!}{=} v_L \\
&\Rightarrow c_2= -\frac 1 8 v_L^2 + \frac 3 8 \rho_L^2 - \frac 1 4 \rho_L v_L +v_L\\
v(\xi)&=-\frac 1 8 \xi^2 + \frac 1 4 (v_L+\rho_L)\xi -\frac 1 8 v_L^2 + \frac 3 8 \rho_L^2 - \frac 1 4 \rho_L v_L +v_L
\end{align*}
\end{document}