Constructing a slope field for a differential equation

Differential equations are any equation which involves a derivative of some kind, usually $\frac{dy}{dx}$ . While we will learn a number of methods to solve such equations, we can also use a slope field to produce a graphical representation of a differential equation and our solution to it.

A slope field comprises a grid of slope lines, which are short lines which have the gradient described by the differential equation at the point where the line is drawn.

For example, suppose we are considering the differential equation

$\frac{dy}{dx} = -xy$

In this example, the gradient depends on both the $x$ and $y$ coordinates. For example, at the point $(1,2)$ , differential equation tells us that

$\frac{dy}{dx} = -(1)(2) = -2$

And so it will be a steeply downward-sloping line at this point. We will later see that the slope field looks like

The completed slope field

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