Mathematics - Differential Equation Question with Solution | TestHub

$\left(x^2{\sin }^{2}y-y^2\cos x\right)dx+\left(x^3\cos y{\sin }^{2}y-2y\sin x\right)dy=0$

$\begin{array}{rl}& \frac{dy}{dx}=\frac{y^2\cos x-x^2{\sin }^{3}y}{x^3\cos y{\sin }^2-2y\sin x}\\ & \left(x^3\cos y{\sin }^{2}y-2y\sin x\right)dy=\left(y^2\cos x-x^2{\sin }^{3}y\right)dx=0\\ & \left(\frac{x^3}{3}d{\sin }^{3}y-\sin d{y}^2\right)-{\sin }^{3}yd\left(\frac{x^3}{3}\right)+y^{2}d\sin x=0\\ & \frac{x^3}{3}d{\sin }^{2}y+{\sin }^{3}yd\left(\frac{x^3}{3}\right)-\left(\sin d{y}^2+y^{2}d\sin x\right)\\ & d\left(\frac{x^3}{3}{\sin }^{3}y\right)-d\left(y^2\sin x\right)=0\end{array}$

$$\frac{x^3}{3}{\sin }^{3}y-y^2\sin x=c$$