In the context of the large buckling of a nonlinearly elastic column under end thrust, this paper treats the design of (constitutive) functions appearing in bifurcation problems so as to produce a prescribed first bifurcating branch (e.g., a branch having a prescribed number of wiggles, which produce a prescribed pattern of hysteresis with snap bucklings in loading-unloading processes). The solution of this design problem also yields a method for determining a constitutive function from a single buckling experiment. A dual variational formulation is used to reduce the design problem to the solution of a linear Volterra integral equation of the first kind with a singular kernel. Effective numerical methods for the solution of such ill-posed equations are described and then applied to some physically interesting examples. Generalizations are discussed.