A differential equation that is a function of x only
a.will produce a slope field with parallel tangents along the diagonal
b.will produce a slope field that does not have rows or columns of parallel tangents
c.will produce a slope field with rows of parallel tangents
d.will produce a slope field with columns of parallel tangents
as with your 2nd post on this, (d) will be the case, since the slope only depends on x.
The correct answer is Option C: A differential equation that is a function of x only will produce a slope field with rows of parallel tangents.
To understand why this is the case, let's first define what a slope field is. A slope field, also known as a direction field, is a graphical representation of the solutions to a differential equation. It shows the slopes of the tangent lines to the solutions at different points in the xy-plane.
When the differential equation is a function of x only, it means that the rate of change of the dependent variable (such as y) with respect to x depends only on the value of x. In other words, the slope of the tangent line to any solution depends only on the x-coordinate.
If we consider a specific point (x, y) in the xy-plane, the slope at that point will be determined by the differential equation. Since the differential equation only depends on x, all points with the same x-coordinate will have the same slope. As a result, the slope field will exhibit rows of parallel tangents.
In contrast, if the differential equation depended only on y (a function of y only), we would see columns of parallel tangents in the slope field. This is because the slope of the tangent lines would only depend on the y-coordinate.
Therefore, the correct answer is option C: A differential equation that is a function of x only will produce a slope field with rows of parallel tangents.