Question

The solution of the differential equation $x\frac{dy}{dx}=-\frac{y}{2}-\frac{\sin 2x}{2y}$ is given by

Answer 1

The differential equation can be rewritten as

$2y\frac{dy}{dx}x=-y^2-\sin 2x$

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

$\Rightarrow \frac{d}{dx}\left(x{y}^2\right)=\frac{d}{dx}\left({\cos }^{2}x\right)$

$\Rightarrow x{y}^2={\cos }^{2}x+c$