Question

The family of curves passing through $(0,0)$ and satisfying the differential equation $\frac{y_2}{y_1}=1$ (where $(y_n=\frac{d^{n}y}{d{x}^n}))$ is

• A. $y=k$
• B. $y=kx$
• C. $y=k(e^x+1)$
• D. $y=k(e^x-1)$

Answer 1

The correct option is A $y=k$

Curves should pass through $(0,0)$
$\frac{y_2}{y_1}=1$

$\frac{d^{2}y}{d{x}^2}=\frac{dy}{dx}$

Veryfying options:
$A:$ $y=k$
$\Rightarrow$ $\frac{dy}{dx}=0$
$\frac{d^{2}y}{dx}=0$

$\frac{d^{2}y}{d{x}^2}=\frac{dy}{dx}$(correct)
$B:$ $y=kx$
$\Rightarrow$ $\frac{dy}{dx}=k$

$\frac{d^{2}y}{d{x}^2}=0$

$\frac{d^{2}y}{d{x}^2}\ne \frac{dy}{dx}$ (wrong)
$C:$ $y=k(e^x+1)$

$\Rightarrow \frac{dy}{dx}=k{e}^x$

$\frac{d^{2}y}{d{x}^2}=k{e}^x$
(But this line doesn't passes through $(0,0)$)
(wrong)
$D:y=K(e^x+1)$ (doesn't passes through $(0,0)$)

Hence the answer is $y=k.$