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DIRECTION FIELDS AND INTEGRAL CURVES

We have seen that the solution of the differential equation is

and since the integration results in the introduction of an arbitrary constant1, this solution represents a one- parameter family of functions called the equation’s integral curves (or solution curves). For example, the differential equation 2 1 has a one-parameter family of solutions given by

2 1

This family consists of parabolas shifted vertically with respect to one another:

Now consider the more general and very important equation

, If is a solution, then

, and therefore

,

However, unless we know , which we do not, we cannot integrate. Nevertheless, equation (2) also represents a family of integral curves of (1) and we can visualize them as follows: if is a solution curve of (1), then at any point , where is defined and differentiable,

,

In other words, , , tells us the slope of the solution curve at , .

If we draw a small line segment of slope , at , , then we know that the local behavior of the solution curve at that that point is like that of a straight line containing the point and of slope , . Figure 1 below illustrates the idea in the particular case where and , 2,1 . Notice that the tangent line has slope 5 at this point.

1 Sometimes we write to stress the fact that a constant is involved.

-2 -1 1 2

-2

2

4

6

8

2

1

0

2 1

2

Figure 1

So although we do not know the exact nature of the solution curve (in red) passing through (2,1), we know how it is changing as it goes through that point; its instantaneous rate of change is 5, that is, it looks like a straight line with slope 5.

With the aid of a computer we can perform this procedure at as many points as we wish and generate a the slope field (also known as a direction field) associated with the differential equation.

In the optional section of these notes, we will show that this field is given by , , .

A slope field looks something like this:

Once many points have been plotted with their corresponding “slope lines”, the profile of the solutions starts to emerge as a “flow of lines” that follow the slope field.

Computer programs such as MATHEMATICA can be used to plot fields as the examples that follow illustrate.

Example 1 The field associated with 2 can be obtained with the command

VectorPlot[{1,2x­y},{x,­a,b},{y,­c,d}] 

where the numbers a, b, c, and d define the desired “window”. The figure below corresponds to the rectangular window 3,3 3,3 .

(2,1)

m = 5

2

1

y = φ(x)

Close-up

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It can be shown that the solution curves of this differential equation are given by the one-parameter family

The line 2 2 seems to separate the solution curves into two distinct families: the ones that lie above the line and the ones that lie below it. We call this a line a separatrix of the integral curves.

Example 2 An even simpler example involves a differential equation in which , involves only the variable x, for then the problem can be integrated and the solution curves explicitly described. Consider the equation

2

In which , 2 . The solution curves are given by 2 . Some members of this family of parabolas are shown in the graph below, as well as the corresponding slope field.

VectorPlot[{1,2x},{x,-2,2},{y,-2,4}]

Isoclines

Drawing a slope field by hand requires a lot of computation but there is another way to study the behavior of solutions qualitatively which is based on a study of the differential equation’s isoclines.

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 1 2

-2

-1

1

2

3

4

-2 -1 0 1 2

-2

-1

0

1

2

3

4

Definition The curves , are called the isoclines of the equation ′ ,

2 2

4

An Isocline can be any kind of plane curve and the name may be misleading. The work “iso” is derived from the Greek for “equal”. Thus, isoclines are plane curves along which the integral curves have equal slope.

The idea of using an equation’s isoclines to analyze the qualitative behavior of solutions is illustrated in the examples below.

Example 3   Determine the isoclines of 2 and use this information to graph some of the solution curves of the differential equation.

Solution The isoclines are found by setting 2 where c is a constant. Thus, the isoclines are straight lines given by

These isoclines are straight lines with slope 2 and y-intercept 0, . For example, if 1, then the corresponding isocline is

2 1

which is a straight line with slope 2 and y-intercept at 0, 1 . What happens along this line? The value of f(x,y) is 1 (the same as that of c). Therefore, if is a solution curve, its slope is also1 at each point on the isoclines:

In the figure, the small red lines represent the slope of the solution curves (blue) as they cross the line 2 1.

The figure below shows the isoclines corresponding to the values 1, 0, 1, and 2. When solution curves intersect these lines, they attain the same slope as the value of c.

The small line segments indicate the slope field, which correspond to the slope of the solution curves. By joining the slope field points one can construct the solution curves (indicated in blue).

Notice that the line corresponding to 2 is the separatrix of this equation. Also observe that 22 is a solution of the differential equation 2 because

2 2 2 2 2 is an identity.

.

c = 1c = 0

c = 2

c = −1

2 1 isocline corresponding to 1 

Integral curves have equal slope 1 along the isocline

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Example 4       The isoclines of the equation   2 are vertical lines given by 2 or /2. Compare the figure below with direction field of example 2.

Example 5   Determine the isoclines of and use this information to graph some of the solution curves of the differential equation.

Solution The isoclines are found by setting , , where c is a constant. Thus, the isoclines are lines of slope 1 given by

Using Mathematica we can plot the direction field (figure on left) and some of the integral curves, which appear in the figure on the right:

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

1 2

0 1

-4 -2 2 4

-3

-2

-1

1

2

3

1 2 3 1 2

2 4 2 4

0

When 1, the isoclines is 1/2. On this line, the solution curves have slope 1/2. The isolcines corresponding to

4, 2, 0, 2, and 4 are shown in the figure to the left.

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Example 6 What are the isoclines of the differential equation 4

Solution The isoclines are given by 4 . Observe that since 4 0 for all , , we can only choose values of c that are non-negative. This means that the solution curves must be always non-decreasing.

The equation 4 represents a family ellipses. For example, if 1 then 4 is an ellipse with x-intercepts at , 0 and y-intercepts at 0, 1 . At each point on this curve, the solution of the differential equation 4 has slope 1. The equation of the ellipses in standard form is

41

and corresponds to a family of concentric ellipses with their major axis along the y-axis (see figure below).

On the ellipse corresponding to 1/2 for example, the solution curves always have slope 1/2. Similarly, they have slope 1 on the ellipse corresponding to 1.

The graph below show the direction field of the equation 4 .

                                                          

Example 7 Graph the direction field of the equation and use its isoclines to determine the equation’s solutions profile.

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

1/2

1 2

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Solution The isoclines are given by which correspond to a family of hyperbolas in the t-u plane. The direction field is given by , and it is illustrated in the figure below:

Observe how the filed seems to align itself with the family of hyperbolas . In order to interpret what the isoclines tell us, consider the three cases 0, 0, and 0 separately. The table below will allow us to organize this information.

0 0 corresponds to the straight lines . On these lines, the solution curves have zero slope.

0 corresponds to hyperbolas with intercepts on the -axis. On these hyperbolas, the solution curves have positive slope.

0 corresponds to hyperbolas with intercepts on the -axis. On these hyperbolas, the solution curves have positive slope.

As one can see, fields can be very useful tools, specially because more often than not, it is impossible to obtain explicit solutions for equations such as the ones of the previous two examples. Later we will develop numerical methods that can be used to find approximations at specific points.

-2 -1 0 1 2

-2

-1

0

1

2

-5 5

-5

5

0 0

0

0

0 0

Typical integral curve