Suppose you plan on going to a picnic on the weekend. You know that it is not a rainy season, so everything would be great. It would be rather unfortunate if your picnic were to be interrupted by some strong winds! Don't you agree? For this reason, it is a good idea to check a weather forecast, because besides telling you if it is going to rain or not, you also get information about the wind. Since wind can blow in different directions, the wind forecasts use graphs known as direction fields.
Here you will learn how to sketch direction fields, also known as slope fields, which are graphical representations of differential equations.
What is the Meaning of a Direction Field?
Graphical representations are very insightful in Calculus. By using a graph, you can relate a mathematical expression to an idea. In this context, mathematical expressions are usually given as equations or inequalities. Take for instance, the function
\[ f(x) = x^2-1,\]
which in order to be graphed, you first need to do the association \( y=f(x)\). This way, you can relate a \(y\) value to each \(x\) value and use a Cartesian Plane to draw all the pairs \( (x, f(x) )\).
Figure 1. Graph of the function \( f(x)=x^2-1\)
Differential equations are no exception when talking about graphs. Of course, you can try to solve a differential equation and get a function as an answer. You can then graph this function, which is the graph of the solution to the differential equation!
However, differential equations still provide some information without having to find a solution. This information is displayed by using direction fields.
Direction fields are more commonly known as slope fields because each segment of the graph represents a slope at a given point.
Direction fields, also known as slope fields, are a graphical representation of first-order differential equations.
While a direction field can be used to picture the solution of a first-order differential equation, please note that the direction field is not the graph of the solution to a differential equation.
Direction Fields and Differential Equations
As mentioned before, direction fields, better known as slope fields, are the graphical representation of first-order differential equations. But you might be wondering, how is it even possible to draw the graph of a differential equation without solving it?
Suppose you are given a differential equation
\[ xy'=2y,\]
which you can rewrite as
\[ y' = \frac{2y}{x}.\]
The basic interpretation of a derivative is that it gives you the slope of a line tangent to a function at a given point. Knowing this, the expression
\[ \frac{2y}{x}\]
is giving you information about the slope at a point \( (x,y) \). By drawing small line segments at each point \( (x,y)\) with the required slope, you can actually picture this differential equation.
Figure 2. Slope field of the differential equation \( xy' = 2y\)
Solving a differential equation requires you to know an integration constant, which will be given depending on the problem. This means that you can have different graphs for the solution to a differential equation.
Figure 3. Solutions to the differential equation \(xy'=2y\) for different values of the integration constant
What happens if you graph some solutions to the differential equation along with the slope field?
Figure 4. Slope field and solutions to the differential equation \( xy'=2y\)
Note that the segments of the slope field are tangent to each solution of the differential equation. This means that you can draw some solutions of a differential equation based on the slope field.
How to Graph Direction Fields
The basic idea for graphing slope fields is to choose a set of points and then use the differential equation to find the slope associated with each point. Then, you can draw a small segment with the respective slope at each of the points you choose. Here are some example segments with different slopes.
Figure 5. Five segments with different slopes
As the absolute value of the slope increases, the segment will be closer to a vertical line. If the slope is positive it will be like a segment of an increasing function, and if it is negative it will resemble a decreasing function.
Figure 6. Two segments whose slopes differ from a sign
Likewise, as the absolute value of the slope decreases, the segment will be closer to a horizontal line.
Figure 7. Two segments whose slopes differ from a sign
There is also the possibility of having completely vertical or horizontal segments. If the slope evaluates to \(0\), then the segment will be completely horizontal. If the slope becomes infinite (positive or negative), then the segment will be completely vertical.
Figure 8. A completely horizontal segment along with a completely horizontal segment
Now that you know which segment to draw depending on the slope, you can proceed to draw some slope fields by choosing some points on the Cartesian plane and drawing segments with the corresponding slope at those points.
Here is an example.
Sketch the slope field of the differential equation
\[ xy' = 2y. \]
Solution:
Begin by isolating \(y'\) from the differential equation. You can achieve this by dividing both sides of the equation by \(x\), that is
\[ \begin{align} \frac{\cancel{x}y'}{\cancel{x}} &= \frac{2y}{x} \\ y' &=\frac{2y}{x}. \end{align} \]
Next, try choosing a few points on each quadrant, as well as some points on both axes.
Figure 9. \(8\) points on the Cartesian plane to be used for the slope field
Note that the point \( (0,0) \) is missing. This is because evaluating the slope at such point will result in an indeterminate form of
\[ \frac{0}{0},\]
so it is better to avoid this point. Now that you have chosen some points, you will need to do plenty of evaluations. You can graph each segment on the go, but if you are still practicing, a table of values will be really helpful.
| \[x\] | \[y\] | \[\frac{2y}{x}\] | \[y'\] |
| \[-2\] | \[2\] | \[\frac{2(2)}{-2}\] | \[-2\] |
| \[-2\] | \[0\] | \[\frac{2(0)}{-2}\] | \[0\] |
| \[-2\] | \[-2\] | \[\frac{2(-2)}{-2}\] | \[2\] |
| \[0\] | \[2\] | \[\frac{2(2)}{0}\] | \[\infty\] |
| \[0\] | \[-2\] | \[\frac{-2}{0}\] | \[-\infty\] |
| \[2\] | \[2\] | \[\frac{2(2)}{2}\] | \[2\] |
| \[2\] | \[0\] | \[\frac{2(0)}{2}\] | \[0\] |
| \[2\] | \[-2\] | \[\frac{2(-2)}{2}\] | \[-2\] |
Finally, you can draw the segments on each point by identifying the slope \(y'\) at each point.
Figure 10. Sketch of the slope field of the differential equation \( xy'=2y\) using \(8\) segments
As usual, you get more information by using more points, but this task would take you some time. However, by inspecting the expression of the slope
\[ y'=\frac{2y}{x}\]
you can note that the segments will become steeper as the value of \(y \) increases. Also, the slope will be positive in the first and third quadrants, while it will be negative in the second and fourth quadrants. You can use this information to improve the sketch of the slope field.
Figure 11. Slope field of the differential equation \( xy' = 2y\)
Sketching a Solution given a Direction Field
Since the direction field of a differential equation gives you a bunch of lines that are tangent to different families of solution curves, you can actually sketch some of these solutions!
Consider the direction field of the differential equation
\[y'=y.\]
Figure 12. Slope field of the differential equation \( y'=y\)
You can note how the lines of the slope field align in a certain pattern. To better identify how these curves behave, imagine you start at the leftmost side of the graph, at any \(y-\)value. As you move towards the right, you will find some slope segments. If these segments have a negative slope, you should move downwards as you move towards the right. Similarly, if the segments have a positive slope, you should move upwards instead.
Figure 13. Sketch of two solution curves for the differential equation \(y'=y\) using its slope field
The solutions to this differential equation are actually a family of exponential functions of the form\[ y(x) = Ae^x,\]
where \(A\) is an integration constant, which can be either positive or negative. This makes perfect sense with the solutions you sketched from the slope field!
Examples of Direction Fields
Here you can take a look at more examples of slope fields.
Sketch the slope field of the differential equation
\[y'=\frac{1}{2}y.\]
Solution:
You are given an isolated form of \(y'\), so you can find the slope of each segment by evaluating the expression
\[\frac{1}{2}y.\]
Note that the above expression does not depend on \(x\), so you do not have to worry about the \(x\) value. This means that you can focus on noting how the slope will be positive for positive values of \(y\), and it will be negative for negative values of \(y\). The lines will become steeper as you move away from the origin, and will be horizontal along all the \(x-\)axis!
Figure 14. Slope field of the differential equation \(y'=\frac{1}{2}y\)
It is also possible that the slope field only depends on the \(x\) value.
Sketch the slope field of the differential equation
\[y'-1=x.\]
Solution:
Begin by isolating \(y'\) from the differential equation. You can achieve this by adding \(1\) to both sides of the equation, obtaining
\[ y'=x+1.\]
This time the above expression does not depend on \(y\). Because of the \(+1\) term, the slope will be positive for \(x>-1\), negative for \(x<-1\), and zero when \(x=-1\). Also, the lines become steeper the further away you are from \(x=-1\).
Figure 15. Slope field of the differential equation \( y'=x+1\)
Direction Fields - Key takeaways
- Direction fields, more commonly known as slope fields, are graphical representations of first-order differential equations.
- A slope field is not the graph of the solution to a differential equation.
- A slope field is made of many segments that represent lines tangent to the different solutions to a differential equation.
- To graph a slope field you need to choose a set of points, find the slope associated with each point using the differential equation, and then graph segments with the corresponding slope.
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