Form differential equation for `Ax+By+C=0`

Options

A. `y_(3)=0`
B. `y_(2)=0`
C. `y_(3)x+y_(2)y+y_(1)=0`
D. `x+yy_(1)+y_(2)=0`

Answer

To form the differential equation for the line represented by the equation \( Ax + By + C = 0 \), we can follow these steps:

### Step 1: Start with the given equation
We have the equation:
\[ Ax + By + C = 0 \]

### Step 2: Differentiate the equation with respect to \( x \)
We differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(Ax + By + C) = 0 \]
Using the rules of differentiation, we get:
\[ A + B\frac{dy}{dx} = 0 \]

### Step 3: Solve for \( \frac{dy}{dx} \)
Rearranging the equation gives us:
\[ B\frac{dy}{dx} = -A \]
Thus, we can express \( \frac{dy}{dx} \) as:
\[ \frac{dy}{dx} = -\frac{A}{B} \]

### Step 4: Differentiate again
Now, we differentiate \( \frac{dy}{dx} \) with respect to \( x \):
\[ \frac{d^2y}{dx^2} = 0 \]
This indicates that the second derivative of \( y \) with respect to \( x \) is zero, which means that the slope \( \frac{dy}{dx} \) is constant.

### Step 5: Write the final form of the differential equation
The final form of the differential equation representing the original line equation \( Ax + By + C = 0 \) is:
\[ \frac{d^2y}{dx^2} = 0 \]

### Summary
The differential equation formed from the line \( Ax + By + C = 0 \) is:
\[ \frac{d^2y}{dx^2} = 0 \]