Although technically not a numerical method, the study of slope fields enables you to obtain a rough sketch of a solution without actually solving the differential equation. In addition, understanding slope fields will help you with the understanding of numerical methods.
The fact that a differential equation gives you the slope for different values, allows you to construct what is called a slope field. Given the equation dy/dx = y, you know that, for any chosen point in the plane, the slope is equal to the value of the y - coordinate at that point. For example if you look at the point (2,3) the slope is 3. If you want the slope at the point (5,8), you know it is 8, because the equation tells you this.
If you were unable to solve the above equation, which you know is not the case, you could theoretically find a graphical solution by looking at these slopes. To do this, you construct a small lineal element, with a slope defined by the equation, at several equally spaced points in the plane. If these lineal elements are close enough together, so that they touched each other, they would create a polygonal curve that would be very close to the curve representing the solution to the equation.
You can construct slope fields by hand and until the introduction of the computer, this is exactly what you would do. The diagram below shows a computer generated slope field for the differential equation dy/dx = y. It was produced using Maple V. Although LiveMath cannot generate slope field diagrams at this time (without a lot of programming, that is), you can use it to help in the generation of isoclines. Isoclines, which are discussed next, are used to construct slope fields by hand.
Maple V has taken equally spaced points and calculated the slope according to the equation dy/dx = y. At each point, it draws a small line with that particular slope. As you can see, the diagram has a pattern which suggests solution curves for the differential equation.
If you were to take a pen and draw a curve, it is evident that the lineal elements would direct you along a solution, hence the name some texts give for this diagram, a Direction Field.
The actual solution to this equation is obtained by using Separation of
Variables and is given as 
. If c=1,
the curve would cross the y-axis at 1 and the curve would look
like the one on the left. The diagram on the right shows several
solution curves as c takes on different values.
To reproduce the diagrams above by hand looks like a daunting task, and it is, if each lineal element is drawn independently. To help draw these lineal elements, there is a method which can help you. Notice above that the slope of each element is the same for single values of y. An isocline is a line which defines equal slopes, and for the equation above is merely F(x,y)=c or y=c. In other words, you can construct lines for different constant values of y in which all the slopes are going to be the same. Below is an example of one of these lines and some lineal elements drawn on it.
This example can be somewhat misleading because of its simplicity.
Not all equations will have horizontal isoclines. In fact most
will not, and in many cases they will not be lines at all. Depending
upon the equation, they may be curves as with the equation dy/dx=
, who's isoclines are concentric circles.
The key here is to set up the differential equation equal to a constant and, if possible, solve for y. The resulting relation will define the isocline. Keep in mind that many times the equation for the isocline will be more difficult to solve than the differential equation itself. In these cases you will have to use other means to find the solution. Let's look at how LiveMath can help you generate isoclines.
The differential equation for this example will use another simple equation which is very similar to the one found in Example 17.
This equation has the solution on the right, above, and was solved with the method used to solve linear equations.
The actual lineal elements are now drawn by using the value of k for each line to define the slopes along that line. A completed graph is shown below.
The following are two Maple V graphs. The first displays a completed slope field for this equation. The second has several solution curves added.
Although slope fields are of little value in determining specific values of a solution at individual points, they can be helpful in establishing the existence of, and the approximate location of, a solution curve. For example, using the equation in the last example, dy/dx = x+y, assume the initial value y(0) = 0. If you wanted to know the value of y(2), you could draw the approximate solution curve and observe that the answer would be close to 4. The solution curve is pretty good, but the value found is only a rough approximation.
When the isoclines are straight lines, in some cases one of these lines can be a solution curve itself. To determine this you merely take the slope of the general isocline equation and use it as the chosen constant. Using the last example again, the equation below demonstrates this.
This solution curve (a straight line) can be observed in the graph above.
First Order Ordinary
Expl #19 Isoclines
