# Video: Deciding Which Curve Represents a Specific Solution to a Differential Equation

The slope field for a particular differential equation is shown. Which curve represents a specific solution to the differential equation? [A] Figure A [B] Figure B [C] Figure C [D] Figure D [E] Figure E

03:12

### 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).