The differential equation of the family of curves $y=e^x(A\,cosx+B\,sinx)$, where A and B are arbitrary constants, is:

$\frac{d^2y}{dx^2}+2\frac{dy}{dx}+2y=0$ $\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0$ $\frac{d^2y}{dx^2}-2\frac{dy}{dx}-2y=0$ None of these

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

$\frac{dy}{dx}=e^x(A\,cosx+B\,sinx)+e^x(-A\,sinx+B\,cosx)=y+e^x(-A\,sinx+B\,cosx)$ $⇒\frac{d^2y}{dx^2}=\frac{dy}{dx}-y+(\frac{dy}{dx}-y)⇒\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0$