Question

If $y=e^{-x}cos2x$ then which of the following differential equation 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

The correct option is A $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+5y=0$

Given, $y=e^{-x}cos2x$ ...(i)

$\therefore \frac{dy}{dx}=-e^{-x}\left(sin2x\right)2+cos2x.e^{-x}(-1)$

$\Rightarrow \frac{dy}{dx}=-2sin2x\cdot e^{-x}-y$

$\Rightarrow \frac{dy}{dx}+y=-2sin2x\cdot e^{-x}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=-2[sin2x\cdot e^{-x}(-1)+e^{-x}2cos2x]$

$\Rightarrow \frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=-\frac{dy}{dx}-y-4y$

$\Rightarrow \frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+5y=0$