# If $y=e^{-x}(A\cos x+B\sin x)$, then $y$ is a solution of

\begin{array}{1 1}(A)\;\frac{d^2y}{dx^2}-2\frac{dy}{dx}=0 & (B)\;\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\\(C)\;\frac{d^2y}{dx^2}+2\frac{dy}{dx}+2y=0 & (D)\;\frac{d^2y}{dx^2}+2y=0\end{array}

Toolbox:
• The general solution of a differential equation is a relation between dependent and independent variables having $'n'$ arbitrary constants.
• A general solution may have moer than one form, but the arbitrary constants must be the same
• $\large\frac{d}{dx}$$(uv)=\large\frac{du}{dv}$$.v+\large\frac{dv}{dx}$$.u Given y=e^{-x}(A \cos x +B \sin x) Let us expand the above equation as y=e^{-x} \;A \cos x +e^{-x} \;B \sin x Consider e^{-x}.A \cos x This in the product form , so let us apply product rule to differentiate this, \large\frac{d}{dx}$$(uv)=\large\frac{d}{dx}$$(u).v+\large \frac{d}{dx}$$(v).u$
Let $u=e^{-x}$ then $\large\frac{d}{dx}$$(u)=-e^{-x} Let v=A \cos x then \large\frac{d}{dx}$$(v)=-A \sin x$
Hence $\large\frac{d}{dx}$$(uv)=-e^{-x}.A \cos x+(-A \sin x).e^{-x} =-e^{-x}[A \cos x+A \sin x] Similarly, e^{-x}.B \sin x=-e^{-x}[B \sin x-B \cos x] Therefore \large\frac{dy}{dx}$$=-e^{-x}[A \cos x+A \sin x+B \sin x-B \cos x]$
$=-e^{-x}[A \cos x+B \sin x]-e^{-x}[A \sin x-B \cos x]$
But $e^{-x}[A \cos x+B \sin x]=y$
Hence $\large\frac{dy}{dx}$$=-y.-e^{-x}[A \sin x-B \cos x] Now differentiating w.r.t x we get, \large\frac{d^2y}{dx^2}=-\frac{dy}{dx}$$-[+e^{-x}[A \cos x+A \sin x+B \sin x+B \cos x]]$
$=-\large\frac{dy}{dx}$$-[e^{-x}(A \cos x+B \sin x)]+[e^{-x}(-A \sin x+B \cos x)] \large\frac{d^2y}{dx^2}=\large\frac{-dy}{dx}-\bigg[\large\frac{dy}{dx}-y\bigg]-y$$\qquad => e^x[-A \sin x+B \cos x=\large\frac{dy}{dx}-y]$
$\large\frac{d^2y}{dx^2}$$+2\large\frac{dy}{dx}$$+2y=0$
Hence the correct option is $C$