The nonlinear two-point boundary value problem \(u''+\lambda^ 2e^ u=0(\lambda >0),u(0)=0=u(1)\) which has a secondary bifurcation point is solved by an analytic numerical method. The problem was first transformed analytically into finding roots of the transcendental equation \(\frac{2\sqrt{2}}{t} \ln (t+\sqrt{t^ 2-1})=\lambda (t>1).\) Then the turning point was obtained numerically, \(\lambda_ c=1.87452030182\). And the following three cases are all considered: the problem has two solutions, one solution, or no solution, when \(\lambda <\lambda_ c\), \(\lambda =\lambda_ c\), or \(\lambda >\lambda_ c\), respectively. The steps for solving the problem and some results of numerical experiments are given in the paper.