Question

If $y=e^{-x}\cos 2x$ 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

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

$y=e^{-x}\cos 2x$
Differentiating both sides
$\frac{dy}{dx}=\cos 2x(-e^{-x})+e^{-x}(-2\sin 2x)$
$\Rightarrow \frac{dy}{dx}=-y-2e^{-x}\sin 2x\cdots \left(1\right)$
Again, differentiate the above equation
$\frac{d^{2}y}{d{x}^2}=-\frac{dy}{dx}-[4e^{-x}\cos 2x-2e^{-x}\sin 2x]$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{dy}{dx}-[4y-2e^{-x}\sin 2x]$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{dy}{dx}-[5y-y-2e^{-x}\sin 2x]$

Substitute the value in the equation $\left(1\right)$
$\frac{d^{2}y}{d{x}^2}=-\frac{dy}{dx}-[5y+\frac{dy}{dx}]$
$\Rightarrow \frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+5y=0$