Slope Fields - Expii

Slope fields allow us to visualize the solution curves \(y = f(x)\) of a first-order ordinary differential equation \(y' = F(x,y)\). Specifically, at each point \((x,y)\) on the plane, we can draw or visualize a short segment with slope \(dy/dx = F(x,y)\). Why is this useful? Well, if \(y = f(x)\) is a solution to \(y' = F(x,y)\), then the segments drawn are all tangent to the graph of the curve \(y = f(x)\). By the way, it turns out that no two solution curves intersect: do you see why?.