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

Options

A. `(d^(2)y)/(dx^(2))=0`
B. `(d^(2)x)/(dy^(2))=0`
C. `(dy)/(dx)=0`
D. `(dx)/(dy)=0`

Explanation

To find the differential equation of all non-vertical lines in a plane, we start with the general equation of a line. The equation of a non-vertical line can be expressed in the slope-intercept form as:

\[ y = mx + c \]

where \( m \) is the slope of the line and \( c \) is the y-intercept. Since we are looking for the differential equation, we need to eliminate the arbitrary constants \( m \) and \( c \).

### Step 1: Differentiate the equation with respect to \( x \)

Differentiating both sides with respect to \( x \):

\[ \frac{dy}{dx} = m \]

Here, \( \frac{dy}{dx} \) represents the slope of the line, which is equal to \( m \).

### Step 2: Differentiate again to eliminate \( m \)

Now, we differentiate \( \frac{dy}{dx} = m \) with respect to \( x \):

\[ \frac{d^2y}{dx^2} = 0 \]

This indicates that the slope \( m \) is constant, which is characteristic of a straight line.

### Conclusion

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

\[ \frac{d^2y}{dx^2} = 0 \]