Slope field

The solutions of a first-order differential equation of a scalar function y(x) can be drawn in a 2-dimensional space with the x in horizontal and y in vertical direction. Possible solutions are functions y(x) drawn as solid curves. Sometimes it is too cumbersome solving the differential equation analytically. Then one can still draw the tangents of the function curves e.g on a regular grid. The tangents are touching the functions at the grid points. However, the direction field is rather agnostic about chaotic aspects of the differential equation.

The slope field can be defined for the following type of differential equations

which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution (integral curve) at each point (x, y) as a function of the point coordinates.

It can be viewed as a creative way to plot a real-valued function of two real variables ${\displaystyle f(x,y)}$ as a planar picture. Specifically, for a given pair ${\displaystyle x,y}$, a vector with the components ${\displaystyle [1,f(x,y)]}$ is drawn at the point ${\displaystyle x,y}$ on the ${\displaystyle x,y}$-plane. Sometimes, the vector ${\displaystyle [1,f(x,y)]}$ is normalized to make the plot better looking for a human eye. A set of pairs ${\displaystyle x,y}$ making a rectangular grid is typically used for the drawing.

...
Wikipedia