If your differential equation y' = f(x,y) satisfies certain regularity conditions, then you're guaranteed a unique solution, which means, among other things, that no two integral curves will intersect. With slope fields, if you're asked to graph a solution of the DE on it, you start at the initial condition, say y(-0.5) = 1, and draw a curve such that the slopes in the slope field are always tangent to your curve. See here for an example. You're given two different initial conditions; if your slope field doesn't have any "intersections" in it, then you should get two separate integral curves (solution curves) for the corresponding DE. Does that make sense?