\begin{array}{l}
 z = y' \Rightarrow z' = y'' \\ 
 a_0 \frac{{dz}}{{dt}} = \sqrt {1 + z^2 } \Rightarrow a_0 \frac{{dz}}{{\sqrt {1 + z^2 } }} = dt \Rightarrow \\ 
 \end{array}