|UBC Calculus Online Course Notes|
You may remember that we have studied the differential equation
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
we observe that it is making a statement about the slope of a tangent line to a curve . 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
This tells us that
if we were to look at a solution that passes through the
at any time
, then the slope of the
solution at that point would have to be
(and similarly for the other values).
We notice in particular that this is independent of the time at which
we look at the solution.
For your consideration:
A second Example
To make the idea of the direction field clearer still, we will look at a second example. Consider the equation
We actually know that the solutions of this differential equation are of the form
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 to its slope . In this case, we would obtain
Now, at each of the
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
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 value will be "guided" towards the t axis, i.e. towards the value .
For your consideration:
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.