Question

If $y=e^{-x}cos2x$ then which of the following differential equations is satisfied ?

• A. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+5y=0$
• B. $\frac{d^{2}y}{d{x}^2}+5\frac{dy}{dx}+2y=0$
• C. $\frac{d^{2}y}{d{x}^2}-5\frac{dy}{dx}-2y=0$
• D. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}-5y=0$

Answer 1

Answer: (A)

Given, $y=e^{-x}\cos 2x$ ...(i)
$\therefore \frac{dy}{dx}=e^{-x}(-\sin 2x)2+\cos 2x\cdot e^{-x}(-1)$
$\Rightarrow \frac{dy}{dx}=-2\sin 2x\cdot e^{-x}-y$
$\Rightarrow \frac{dy}{dx}+y=-2\sin 2x\cdot e^{-x}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}$
$=-2[\sin 2x\cdot e^{-x}(-1)$
$+e^{-x}2\cos 2x]$
$=2\sin 2x\cdot e^{-x}-4y$ [from Eq. (i)]
$\Rightarrow \frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=-\frac{dy}{dx}-y-4y$
$\Rightarrow \frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+5y=0$