Dissections, transgressions, and perilous paths

A model of the mass transfer from a bubble rising through a liquid has been shown to have equations of the form dxdt=A(x,y)B(x,y), dydt=Г(x,y)Δ(x,y). This is one of a number of dissections of the equation dydx=BГAΔ, all of which have the same solution paths. These equations will clearly have singular behavior on the curves B = 0 and Δ = 0, but the boundary between physically realistic and physically unrealistic values of x and y must be transgressed to reach this interesting behavior. It is shown that some of the dissections give families of singular trajectories which owe their perilous “stability” to the opposition of two infinite derivatives.