Direction Fields

One important tool that can help us better understand the behaviour of a differential equation without actually solving it is known as a direction field which we will define below.

Definition: The Direction Field or Slope Field of a differential equation is a rectangular grid of tangential line segments that pass through partial solutions at that point on the grid.

Direction fields are best understood with an example. Consider the following differential equation $\frac{dy}{dx} = 2y - 100$ where $y = f(x)$. We will now construct a direction field for this differential equation by plugging in various values of $y$ and seeing the behaviour of the slopes $\frac{dy}{dx}$ as $x$ varies.

We first notice that $y = 50$ is a solution to the differential equation above, and plugging in $y = 50$ into our differential equation gives us $\frac{dy}{dx} = 0$. So when $y = 50$, the slopes of the tangents of $y$ are horizontal. To start off our direction field, we will graph this line, $y = 50$:

Now if $y > 50$, say, $y = 55$, then we have that $\frac{dy}{dx} = 110 - 100 = 10 > 0$. In particular, if we choose $y > 50$, then $\frac{dy}{dx} > 0$ and so the slopes of $y$ will be positive. In fact as $y \to \infty$, the slopes of $y$ get steeper and steeper (positively) since:

(1)

\begin{align} \quad \lim_{y \to \infty} \frac{dy}{dx} = \lim_{y \to \infty} 2y - 100 = \infty \end{align}

Now if $y < 50$, say, $y = 45$, then notice we have that $\frac{dy}{dx} = 90 - 100 = -10 < 0$. In this case, if we choose $y < 50$, then $\frac{dy}{dx} < 0$ and so the slopes of $y$ will be negative. In fact as $y \to -\infty$, the slopes of $y$ get steeper and steeper (negatively) since:

(2)

\begin{align} \quad \lim_{y \to -\infty} \frac{dy}{dx} = \lim_{y \to -\infty} 2y - 100 = -\infty \end{align}

Below is a graph of the directional field of the differential equation $\frac{dy}{dx} = 2y - 100$: