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

A. $\frac{d y}{d x}=\frac{x-y}{y-1}$
B. $\frac{d y}{d x}=\frac{x+1}{y+1}$
C. $(x+1) \frac{d y}{d x}-y=0$
D. $(x+1) \frac{d y}{d x}+y=0$
E. $\frac{d y}{d x}-\frac{x+1}{y+1}=0$

Choose the correct answer from the given below:

C only

The correct answer is Option (3) - C only

let eq. be $\frac{x}{a}+\frac{y}{b}=1$   ...(1)

$\frac{dy}{dx}=b$

differentiating (1) w.r.t. x

$\frac{1}{a}+\frac{1}{b}\frac{dy}{dx}=0$

$\frac{1}{a}++1=0⇒a=-1$

from (1) $-x+\frac{y}{b}=1$

so $y=b(x+1)$

$y=\frac{dy}{dx}(x+1)$

$\frac{dy}{dx}(x+1)-y=0$