Question

If $y^2=a{e}^{-2x}+\left(2/5\right)(cosx-2sinx)$ then
$y\frac{dy}{dx}+y^2+sinx$ is equal to

• A. -1
• B. 1
• C. 0
• D. none of these

Answer 1

The correct option is C 0

$\begin{array}{c}{y}^2=a{e}^{-2x}+\frac{2}{5}\left(\cos x-2\sin x\right)\\ 2y\frac{dy}{dx}=-2a{e}^{-2x}+\frac{2}{5}\left[-\sin x-2\cos x\right]\\ y\frac{dy}{dx}=-a{e}^{-2x}+\frac{1}{5}\left[-\sin x-2\cos x\right]\\ y\frac{dy}{dx}+y^2+\sin x\\ =-a{e}^{-2}-\frac{1}{5}\left[\sin x+2\cos x\right]+a{e}^{-2x}+\frac{2}{5}\left(\cos x-2\sin x\right)+\sin x\\ =0\\ Hence,\\ optionCiscorrectanswer.\end{array}$