Video Transcript
The slope field for a particular
differential equation is shown. Which curve represents a specific
solution to the differential equation?
We have five possible answers
here. And as we go through, I’ll make
each slope feel bigger so we can take a proper look. Recall that a slope field shows
possible solutions to a differential equation. Each line segment represents the
slope 𝑦 prime or d𝑦 by d𝑥 of a particular solution at a point. Then any curve which follows the
flow of these line segments is a solution to the differential equation.
We’ve got five lines or curves all
shown on the same slope field, and we’ve got to decide which one represents a
specific solution to the differential equation.
Let’s start by taking a look at the
possible solution (A). We need to check whether this line
follows the general flow of the line segments in the slope field. If we look at the line segments in
the slope field around this possible solution, we see that many of them are actually
vertical. Some of them are even perpendicular
to this possible solution line. Therefore, it appears that this
line does not follow the general flow of the line segments of the slope field. Therefore, this line cannot be a
possible solution to the differential equation represented by this slope field.
Let’s have a look at possible
solution (B). Again, we look at the line segments
around the slope to see whether the slope follows the general flow of the line
segments. Actually, we find that this slope
does not follow the general flow of the line segments. Many of the line segments around
the slope are going in a different direction to the slope. So this is not a solution to the
differential equation.
Let’s have a look at solution curve
(C). Again, we’ll look and see whether
this curve follows the general flow of the line segments. If we look carefully at the line
segments around the slope, we can see that this slope does follow the general flow
of the line segments. Therefore, this curve is a solution
to the differential equation demonstrated by this slope field.
Let’s still go ahead and check (D)
and (E). Again, let’s look at the line
segments around these two curves. Whilst one or two of the line
segments do fit this curve, we generally find that many of them are going in the
opposite direction to the curve shown. So as this doesn’t follow the
general flow of the line segments, this is not a solution to the differential
equation.
So let’s finally check possible
solution (E). Let’s look at the line segments
around this curve. Whilst we see that in the bottom
part of the graph the curve sort of follows the direction of the line segments,
although it’s not as vertical as the line segments, the top part of the graph we
find that the line segments go in a different direction to the curve. Therefore, we can say that this is
not a solution to the differential equation represented by this slope field.
Therefore, we can conclude that the
curve which represents a specific solution to the differential equation represented
by this slope field is (C).
