Summary: We study the number and properties of the solutions of the equation \(\varepsilon u_{xx}=u(u-1) (u-\lambda)\), \(u(0)=0\), \(u(1)= \mu\), \(\lambda\in(0,1)\), depending on the values of \(\varepsilon>0\) and \(\mu\geq 0\). It is found that all solutions are either increasing or have only one critical value, a maximum. The positive quadrant of the \((\varepsilon,\mu)\)-plane is cut by bifurcation curves into regions with: a) one solution only (of the first type); b) one solution of the first type and even number of solutions of the second type (at least two); c) no solutions of the first type and one solution of the second type; d) no solutions of the first type and three or more (odd number) solutions of the second type. The geometry of the regions is investigated in detail.