1.2 Direction Fields

Although having an explicit formula for the solution of a differential equation is useful for understanding the nature of the solution, determining where it increases or decreases, and identifying its maximum or minimum values, finding such a formula is often impossible for most real-world differential equations. Consequently, alternative methods are employed to gain insights into these questions. One effective approach for visualizing the solution of a first-order differential equation is to create a direction field for the equation. This method provides a graphical representation of the solution’s behavior without requiring an explicit formula.

We assume that the first-order differential equation [asciimath]y'=f(x,y)[/asciimath] has solutions. For this equation, function [asciimath]f(x,y)[/asciimath] gives the slope of the solution curve at any point $[asciimath](x,y)[/asciimath]$ in the XY-plane. In a direction field, these slopes are represented by small line segments or arrows, drawn at a selection of points in the plane. Each segment has a slope equal to the value of $[asciimath]f(x,y)[/asciimath]$at that point.

Figure 1.2.1 Direction field for and solutions to [asciimath]y'=x+y[/asciimath]

The arrows in the direction fields represent tangents to the actual solutions of the differential equations. We can use these arrows as guides to sketch the graphs of the solutions to the differential equation, providing a visual representation of how the solutions behave. By following these arrows, we can visually trace the trajectory of a solution over time, which can indicate its long-term behavior.