Abraham (1987) Simple Job Market Matching Model
Note that function $p$ actually depends upon a single variable $u/v$ and I prefer avoiding the abuse of notation 
(source of confusion) and write $p(u/v) = P(u,v)$. This implies:

\begin{equation}
p'(u/v) = P_u(u,v)v = P_v(u,v) \cdot (-v^2/u). 
\end{equation}

So if $P(u,v)v = s$ total differentiation along $ds=0$ yields

\begin{equation}
P_u(u,v)vdu + (P+P_v(u,v)v)dv = 0, 
\end{equation}

or equivalently 

\begin{equation}
\frac{dv}{du} = - \frac{P_uv}{P+P_v v}. 
\end{equation}

This result is still slighly different from those given in your question... Who can help further?