The differential equation which represents the family of curves $y =C_1e^{C_2x}$, where $C_1$ and $C_2$ are arbitrary constants, is

$yy"=(y')^2$

The correct answer is option (4) : $yy"=(y')^2$

The equation of the family of curves is

$y=C_1e^{C_2x}⇒log\, y = log\, C_1+C_2x$

Differentiating w.r. to x, we get

$\frac{1}{y}\frac{dy}{dx}=C_2$

Differentiating w.r. to x, we get

$\frac{1}{y}\frac{d^Y}{dx^2}-\frac{1}{y}\left(\frac{dy}{dx}\right)^2 = 0 $

$⇒y\frac{d^y}{dx^2}=\left(\frac{dy}{dx}\right)^2$ or, $yy' = (y')^2$