Differential Equations

We need to determine the differential equation that represents all non-vertical lines in a plane. A line in the plane can be expressed in the slope-intercept form as , where is the slope and is the y-intercept. For any line, the slope is defined by the derivative , which signifies that the rate of change of with respect to is constant.

To find the differential equation, consider the derivative of :

The second derivative, , represents the rate of change of the slope. For a straight line, this rate of change is zero since the slope is constant. Therefore, for a non-vertical line, we have:

This equation indicates that the second derivative of with respect to is zero, meaning that the slope (first derivative) of the line does not change as changes. Thus, the differential equation of all non-vertical lines in a plane is .