I'm not sure I understand your question clearly.
But generally speaking, Picard's theorem asserts that a differential equation whose coefficients are continuous has only one solution through any given point.
The brachistochrone is a second order linear equation, and thus, a "point" in the equations definition area consists of values for t, y(t) and y'(t).
This can also be separated two two different boundary conditions, e.g. two tuples (t_0,y(t_0)) and (t_1,y'(t_1)).
The representation of the brachistochrone as a linear equation asserts a certain location at a given time, and a maximal speed (which is the derivative of location, roughly speaking) at another time, which give initial conditions on both y and y'.