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

Answer: (A)

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

$\therefore y_1=-y-2e^{-x}\sin 2x.....\left(1\right)$
$\therefore y_2=-y_1+2e^{-x}\sin 2x-4e^{-x}\cos 2x$
$=-y_1-y-y_1-4y$
$\Rightarrow y_2+2y_1+5y=0$