Question

Determinar si la ecuacion es solucion o no de la ecuacion diferencial dada
$\sin \:y\left(\frac{dy}{dx}\right)\:+\sin \:x\cdot \sin \:y\:=\sin \:x$

Answer 1

$\sin y\left(\dfrac{dy}{dx}\right)+\sin x\cdot \sin y=\sin x\\ \\ \\ \sin y\left(\dfrac{dy}{dx}\right)=\sin x-\sin x\cdot\sin y \\ \\ \\ \sin y\left(\dfrac{dy}{dx}\right)=\sin x\cdot (1-\sin y) \\ \\ \\ \dfrac{\sin y}{1-\sin y}~dy= \sin x~dx\\ \\ \\ \text{Integrando...}\\ \\ \displaystyle \int\dfrac{\sin y}{1-\sin y}~dy= \int\sin x~dx\\ \\ \\ \tan\left(\dfrac{y}{2}+\dfrac{\pi}{4}\right)=-\cos x+C\\ \\ \\ \boxed{y=2\arctan(C-\cos x)-\dfrac{\pi}{2}}$