This paper deals with the solvability of two-point boundary value problems for the nonlinear ODE \( u^{''} = \lambda sh(\mu u) \) , \( \lambda > 0 \) , \( \mu > 0 \) equipped with two point boundary data for \( t= 0 \) , \( t=1 \) , respectively for \( t = - \infty \) and \( t = 0 \) . In the first case we consider the Troesch’s boundary value problem. The above mentioned problems have several applications into mathematical physics. We construct an exact solution of the Troesch’s problem expressing it by the Jacobi’s elliptic functions sn, cn in a linear fractional form. The solution of the second boundary value problem is written by elementary functions and its two approximate solutions are found for \( t \le -1 \) and for \( \lambda = \mu \) sufficiently large. For fixed values of \( \mu = \lambda > 0 \) the behavior of u(t) near \( t = 0 \) is given too. It is interesting to note that the solutions of the ODE under consideration blows up for finite time and consequently are not globally defined on \( \textbf{R}^{1} \) .