Summary: §1.2: DIRECTION FIELDS
In Math 3B, we think a lot about finding an unknown function F(t)
whose derivative is known, some given function f(t). One approach
is geometric. At each point in the plane, we know the slope of the
function F(t), so we can plot a little segment with that slope, and
cover the whole plane. This gives a picture of what the graph of
f(t) should look like. The top of Figure 1 shows this for f(t) = t.
Convince yourself of this: the slopes are large when the t coordinate
is large and small when the t coordinate is small. The slopes do not
depend on the y coordinate at all.
We can see solutions by following the slopes. They are parabolas
just as we expect, since F(t) = t2
/2 + a constant. This idea occurs in
Math 3B but is not emphasized.
In this course, the derivative of the unknown function depends
on both the t and y coordinates. The same idea as above gives the
DIRECTION FIELD or SLOPE FIELD for the differential equation. In
general it does not have the symmetry under vertical translation that
the 3B examples had, so the problems are harder now. The bottom
of Figure 1 shows the direction field for