Qualitative ideas and direction fields

You may remember that we have studied the differential equation

$\frac{dy}{dt} = y$

and that solutions of this equation were exponential functions of the form

$y(t) = y_o e^t$

We found this relationship between the exponential function and the differential equation above by examining derivatives and noticing a connection between the exponential function and its derivative. But for many differential equations that crop up in scientific applications, it is not very easy to see such relationships. It is thus useful to build some tools for understanding solutions to differential equations when they are not known or easily found in advance. Below we will build some tools for doing this.

Things to notice

If we look at the differential equation

$\frac{dy}{dt} = y$

we observe that it is making a statement about the slope $ dy/dt $ of a tangent line to a curve $ y(t) $ . Indeed, this equation is telling us that the slope is numerically equal to the value of the function! For example, we might make a table that links values of the function with values of its slope

Value of	0	1	2
Value of	0	1	2

This tells us that if we were to look at a solution that passes through the point $ y=1 $ at any time $ t $ , then the slope of the solution at that point would have to be $ dy/dt = y = 1 $ , (and similarly for the other values). We notice in particular that this is independent of the time at which we look at the solution.
The differential equation is actually giving us information about a collection of tangent lines ! The slope of the tangent lines is connected to their height in the $ yt $ plane, and does not depend on the value of the independent variable $ t $ .

But what does all this mean? And how can all these tangent lines help us to understand the behaviour of a solution that we do not know in advance?

The direction field

Suppose we draw a collection of tangent lines as shown in the picture below. This collection is often called a direction field since it seems to give us a sense of direction in the $ yt $ plane. Remember that for every starting value (initial value) $ y(0)=y_o $ , there is a specific exponential function $y(t)=y_o e^{t}$ . We have shown just one of these specific curves in blue below. Notice that the curve has an overall shape and direction "guided by" the direction field. Indeed, we could say that

A solution to the differential equation $ dy/dt = y $

has a graph whose tangent line at any point is one of the directions on its direction field.

For your consideration:

(1) By changing the initial value of the solution shown above, experiment with the curve and show that no matter where it starts, it will always follow the direction prescribed by the red lines.
(2) What is the initial slope of the solution that has initial value ?
(3) What would be the initial slope of the curve if the value of was negative?

A second Example

To make the idea of the direction field clearer still, we will look at a second example. Consider the equation $\frac{dy}{dt} = - \frac{1}{2} y$

We actually know that the solutions of this differential equation are of the form

$y(t) = y_o e^{-\frac{1}{2}t}$

But let us "pretend" momentarily that we forgot or never knew this fact. Again, we could let the differential equation guide our qualitative understanding of how the solution behaves. We can make a similar table relating the values of the function $ y $ to its slope $ dy/dt $ . In this case, we would obtain

Value of	0	1	2	3
Value of	0	-1/2	-1	-3/2

Now, at each of the $ y $ values above, (and at some others, not shown in the table) we might again draw a little representation of a line with corresponding slope. The result, when we collect these together in the $ yt $ plane might look something like this:

You might notice that now, all the slopes are negative in the first quadrant, so that any solution starting out with a positive $ y $

value will be "guided" towards the t axis, i.e. towards the value $ y=0 $

For your consideration:

(1) Experiment with various initial values of the solution shown above in blue. Is the blue curve always tangent to the red lines?
(2) What is the initial slope of the solution that has initial value ?
(3) What would be the initial slope of the curve if the value of was negative?

The ideas we have discussed here form the beginnings of an approach called the qualitative analysis of differential equations. Even if we had no idea what exponential functions looked like, or that they were the solutions of the types of differential equations discussed here, the simple ideas connecting differential equation with tangent lines and with a direction field would have certainly conveyed to us the prediction of fast growth or of decay of solutions to such equations.

We will see in the next few pages that these ideas are enormously useful as they can lead us to understand the behaviour of more complicated cases which are not so simple to understand from studying patterns of derivatives.