The visualisation tool below demonstrates the solutions of the ODE

dy/dx = 1/(y-1)^2

graphically. A particular solution is shown, which can be changed by changed the value of the initial condition in the slider.
Blue arrows: The gradient of the solutiuon at that point i.e. the value of dy/dx.
Red line: The line along which dy/dx -> infinity (i.e. the solution gradient is vertical).
Green line: The particular solution corresponding to the initial condition set by the slider value.