Question Video: Finding Where a Differential Equation Equal to Zero Using the Slope Field Graph Mathematics

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.