Expii

# 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?.