$y=A{e}^x+B{e}^{-2x}$

On differentiating, we get

$\frac{dy}{dx}=A{e}^x-2B{e}^{-2x}$

Subtracting both the above equations, we get

$y-\frac{dy}{dx}=3B{e}^{-2x}$

or $y{e}^{2x}-\frac{dy}{dx}{e}^{2x}=3B$

Differentiating above equation, we get

$2y{e}^{2x}+e^{2x}\frac{dy}{dx}-2e^{2x}\frac{dy}{dx}-e^{2x}\frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow 2y-\frac{dy}{dx}-\frac{d^{2}y}{d{x}^2}=0$