First order non linear equation with trigonometric function

Question

I cannot solve and find this equation. Can you help me?

$$2x^2\cdot \bigg(\frac{\cos y}{\sin y}\bigg)\cdot \frac{dy}{dx} = 5x-3\sin y$$

Answer 1

Hint: let $u:=\frac{\sin y}{x}$ so $\frac{du}{u(1-u)}=\frac{3dx}{2x}$ (double-check that), i.e. $\ln|\frac{u}{1-u}|=\frac{3}{2}\ln |x| + C$.

Answer 2

Multiply by $\sin(y)$, set $u=\sin y$, then $$ 2x^2u'=5xu-3u^2 $$ is a Bernoulli-equation. The integrating factor of $$ (u^{-1})'+\frac{5}{2x}(u^{-1})=\frac{3}{2x^2} $$ is $x^{5/2}$, thus with $v(x)=\frac{x^{5/2}}{u(x)}$ one gets $$ v'(x)=\frac32x^{1/2}\implies v=x^{3/2}+C $$