The differential equation which represents the family of curves $y=C_1{e}^{C_{2}x}\text{, where }{C}_1\text{ and }{C}_2$ are arbitrary  constants, is

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

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

Differentiating w.r. to x, we get

$\frac{1}{y}\frac{d^{2}y}{d{x}^2}-\frac{1}{y^2}{\left(\frac{dy}{dx}\right)}^2=0\Rightarrow y\frac{d^{2}y}{d{x}^2}={\left(\frac{dy}{dx}\right)}^2\text{ or, }y{y}^{\prime \prime }={\left(y^{\mathrm{\prime }}\right)}^2$