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

$y y''=(y')^2$

The equation of the family of curves is

$y=C_1 e^{C_2 x} \Rightarrow \log y=\log C_1+C_2 x$

Differentiating w.r. to $x$, we get

$\frac{1}{y} \frac{d y}{d x}=C_2$

Differentiating w.r. to $x$, we get

$\frac{1}{y} \frac{d^2 y}{d x^2}-\frac{1}{y^2}\left(\frac{d y}{d x}\right)^2=0 \Rightarrow y \frac{d^2 y}{d x^2}=\left(\frac{d y}{d x}\right)^2$ or, $y y''=\left(y'\right)^2$