  We consider the connection problem of the second nonlinear differential
equation \begin{equation} \label{eq:1}
  \Phi''(x)=(\Phi'^2(x)-1)\cot\Phi(x)+ \frac{1}{x}(1-\Phi'(x)) \end{equation}
subject to the boundary condition $\Phi(x)=x-ax^2+O(x^3)$ ($a\geq0$) as
$x\to0$. In view of that equation (1) is equivalent to the fifth Painlev\'e
(PV) equation after a M\"obius transformation, we are able to study the
connection problem of equation (1) by investigating the corresponding
connection problem of PV. Our research technique is based on the method of
uniform asymptotics presented by Bassom el at. The monotonically solution on
real axis of equation (1) is obtained, the explicit relation (connection
formula) between the constants in the solution and the real number $a$ is also
obtained. This connection formulas have been established earlier by Suleimanov
via the isomonodromy deformation theory and the WKB method, and recently are
applied for studying level spacing functions.
