Video Transcript
Consider the given slope field
graph. At which point is the differential
equation equal to zero? Point 𝐴, point 𝐵, point 𝐶, point
𝐷, or point 𝐸.
Recall that slope fields show as
possible solutions to differential equations. Each line segment represents the
slope d𝑦 by d𝑥 for a particular solution at that value of 𝑥 and value of 𝑦. We can see that in the slope field,
we have some positive slopes and some negative slopes. One thing to notice about this
particular slope field is that in each column, the line segments have the same
slopes.
For example, if we look at the
point where 𝑥 is equal to four, regardless of the value of 𝑦, all the line
segments at the point 𝑥 equals four are the same. They all have the same slope. So what this means is that the
slopes depend on 𝑥 and not 𝑦. So, from this, we can infer that
d𝑦 by d𝑥 is a function of 𝑥 alone.
Now, we know that where d𝑦 by d𝑥
is equal to zero, we have a horizontal slope. So we’re looking for horizontal
segments in order to see where this differential equation is equal to zero. And if we look carefully, we can
see that horizontal line segments occur where 𝑥 is equal to two. And the only one of our options
which occurs here is the point 𝐴. Therefore, we can conclude that
this differential equation is equal to zero at the point 𝐴 only.
