Slope field

Definition

Let $\Omega$ be an open subset of $\mathbb{R}^2$ and let $f : \Omega \to \mathbb{R}$ be a function. The slope field of the first-order equation $y' = f(x, y)$ is the map that assigns to each point $(x, y) \in \Omega$ the line through $(x, y)$ with slope $f(x, y)$. In a picture, one draws a short segment of this line at each point of a grid in $\Omega$.

Remarks

Let $u$ be a differentiable function on an interval. It is a solution of $y' = f(x, y)$ if and only if its graph is tangent to the slope field at each of its points. For $y' = y$, the segments are horizontal on the $x$-axis and get steeper as $|y|$ grows. For a system $x' = F(x)$ in the plane, the same picture with arrows is a picture of the vector field $F$.

Explanations

L1 human

A slope field shows the direction that a solution must take at each point. You can sketch a solution without a formula: start at a point and follow the small segments. This gives a quick picture of how all the solutions behave.