Question

The differential equation of all non-vertical lines in a plane is

• A. \( \frac{dy^2}{dx^2} = 0 \)
• B. \( \frac{dx}{dy} = 0 \)
• C. \( \frac{dz}{dx} = 0 \)
• D. \( \frac{dz}{dy} = 0 \)

Hint:

For any non-vertical line, the slope \( \frac{dy}{dx} \) is constant, and this is reflected in the differential equation. Vertical lines do not satisfy this equation, as their slope is undefined.

Answer 1

The Correct Option is A

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 \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. For any line, the slope \( m \) is defined by the derivative \( \frac{dy}{dx} = m \), which signifies that the rate of change of \( y \) with respect to \( x \) is constant.

To find the differential equation, consider the derivative of \( \frac{dy}{dx} \):

The second derivative, \( \frac{d^2y}{dx^2} \), 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:

\(\frac{d^2y}{dx^2} = 0\)

This equation indicates that the second derivative of \( y \) with respect to \( x \) is zero, meaning that the slope (first derivative) of the line does not change as \( x \) changes. Thus, the differential equation of all non-vertical lines in a plane is \( \frac{d^2y}{dx^2} = 0 \).

Answer 2

To find the differential equation of all non-vertical lines in a plane, we begin by considering the standard equation of a line: \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

A non-vertical line has a defined slope \(m\), and thus the function \(y(x)\) is differentiable with respect to \(x\). The first derivative \( \frac{dy}{dx} = m \) is a constant since the slope of a line is constant.

To derive the differential equation, we differentiate the slope \( \frac{dy}{dx} \) again with respect to \(x\):

\(\frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dx}(m) = 0\)

Thus, the second derivative \( \frac{d^2y}{dx^2} \) is zero, indicating that the rate of change of the slope is zero.

Hence, the differential equation representing all non-vertical lines is:

\(\frac{d^2y}{dx^2} = 0\)