Video Transcript
A particular solution of the differential equation whose slope field is shown contains the point 𝑃. Which other point may this solution also contain?
We have the slope field of some differential equation. And we’re interested in the particular solution to that differential equation, which goes through the point 𝑃. In particular, we’re interested in what other points — 𝐴, 𝐵, 𝐶, or 𝐷 — the solution may contain. The best way to find out is to sketch the particular solution.
We start at the point 𝑃 and go in the direction suggested by the line segments of the slope field, so initially down and right. As we proceed, we see the line segments start to point more down and right. And as we cross the 𝑥-axis, in fact, they start to point left. As we continue, trying as much as possible to stay in the direction of the line segments, we see that we’re approaching the point 𝐴.
Of course, as this is just a sketch, we may not hit 𝐴 directly. We may have to swerve slightly to get to 𝐴. But we certainly get very close to 𝐴 in a way that we don’t get close to 𝐵, 𝐶, or 𝐷. The particular solution doesn’t start at 𝑃 but goes through it. And so we should sketch the curve in the other direction from 𝑃 as well, just to make sure that we don’t hit 𝐵, 𝐶, or 𝐷 by going in this direction. I think it’s clear that we do not.
And so, with some confidence, we say that our answer is 𝐴. The curve that we drew using the slope field goes through 𝐴 but doesn’t come near the other points 𝐵, 𝐶, or 𝐷. This curve represents the particular solution of the differential equation which contains the point 𝑃. And so we see that this particular solution of the differential equation also contains the point 𝐴.
