Find the differential equation of All-horizontal lines in a plane. All non-vertical lines in a plane.

Explanation

To find the differential equation of all horizontal lines in a plane and all non-vertical lines in a plane, we can follow these steps:

### Step 1: Understanding the Equation of a Line
The general equation of a line in slope-intercept form is given by:
\[ y = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept.

### Step 2: Finding the Differential Equation for Horizontal Lines
For horizontal lines, the slope \( m \) is equal to 0. Thus, the equation simplifies to:
\[ y = c \]
where \( c \) is a constant.

### Step 3: Differentiate the Equation
Differentiating both sides with respect to \( x \):
\[ \frac{dy}{dx} = 0 \]
This indicates that the slope of the line is zero, which is consistent with the definition of horizontal lines.

### Step 4: Second Differentiation
To form the differential equation, we can differentiate again:
\[ \frac{d^2y}{dx^2} = 0 \]
This indicates that the second derivative of \( y \) with respect to \( x \) is zero, confirming that the line is horizontal.

### Conclusion for Horizontal Lines
The required differential equation for all horizontal lines in a plane is:
\[ \frac{d^2y}{dx^2} = 0 \]

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### Step 5: Finding the Differential Equation for Non-Vertical Lines
For non-vertical lines, the general equation remains:
\[ y = mx + c \]
where \( m \) is not equal to infinity.

### Step 6: Differentiate the Equation
Differentiating both sides with respect to \( x \):
\[ \frac{dy}{dx} = m \]
Here, \( m \) is a constant that represents the slope of the line.

### Step 7: Second Differentiation
Differentiating again:
\[ \frac{d^2y}{dx^2} = 0 \]
This indicates that the second derivative of \( y \) with respect to \( x \) is zero, which means the slope \( m \) is constant.

### Conclusion for Non-Vertical Lines
The required differential equation for all non-vertical lines in a plane is:
\[ \frac{d^2y}{dx^2} = 0 \]

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