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\begin{document}

\noindent Solution for assignment 15: \bigskip

The total Energy the is
\begin{eqnarray}
  E &=& \frac{1}{2}\,\dot x^2 + \frac{1}{2}\frac{k}{x^2}\ .
\end{eqnarray}
The initial values imply
\begin{eqnarray}
  E &=& \frac{1}{2}\frac{k}{x_0^2}\ .
\end{eqnarray}
Therefore,
\begin{eqnarray}
  \dot x^2 &=& \frac{k}{m}\,\left(\frac{1}{x_0^2}-\frac{1}{x^2}\right)\\
  \dot x &=& \pm \sqrt{\frac{k}{m\,x_0^2}}\,\frac{\sqrt{x^2-x_0^2}}{x}\ .
\end{eqnarray}
Separation of variables gives 
\begin{eqnarray}
  \pm \sqrt{\frac{k}{m\,x_0^2}}\,\int_0^t dt' &=&
  \pm \sqrt{\frac{k}{m\,x_0^2}}\,t\ =\ \int_{x_0}^x dx'\,
  \frac{x'}{\sqrt{x'^2-x_0^2}}\ =\ \sqrt{x^2-x_0^2}\ .
\end{eqnarray}
Therefore, the solution is 
\begin{eqnarray}
  x(t) &=& \sqrt{\frac{k}{m\,x_0^2}\,t^2 + x_0^2}\ .
\end{eqnarray}
\end{document}
