The differential equation of the family of straight lines whose slope is equal to y - intercept ,is

Options

A. ` (x+1)(dy)/(dx)-y = 0`
B. ` (x+1)(dy)/(dx)+y = 0`
C. ` (dy)/(Dx)=(x+1)/(y-1)`
D. `(dy)/(Dx)=(x+1)/(y+1)`

Explanation

To find the differential equation of the family of straight lines whose slope is equal to the y-intercept, we can follow these steps:

### Step 1: Understand the relationship between slope and y-intercept
The general equation of a straight line can be expressed as:
\[ y = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept. According to the problem, the slope \( m \) is equal to the y-intercept \( c \).

### Step 2: Substitute the relationship into the equation
Since \( m = c \), we can substitute \( m \) with \( c \) in the equation:
\[ y = cx + c \]

### Step 3: Factor out the constant \( c \)
We can factor out \( c \) from the equation:
\[ y = c(x + 1) \]

### Step 4: Differentiate the equation
To find the differential equation, we need to differentiate the equation with respect to \( x \):
\[ \frac{dy}{dx} = c \]

### Step 5: Express \( c \) in terms of \( \frac{dy}{dx} \)
From the differentiation, we have:
\[ c = \frac{dy}{dx} \]

### Step 6: Substitute \( c \) back into the original equation
Now, we substitute \( c \) back into the original equation \( y = c(x + 1) \):
\[ y = \frac{dy}{dx}(x + 1) \]

### Step 7: Rearrange to form the differential equation
This gives us the differential equation:
\[ y = \frac{dy}{dx}(x + 1) \]

### Final Result
Thus, the differential equation of the family of straight lines whose slope is equal to the y-intercept is:
\[ y = \frac{dy}{dx}(x + 1) \]

---