Solucionar 2ydy=-x/dxsinx

$$2 y d y = - \frac { x } { d x } \sin x$$

Resolver para d (solución compleja)

$$\left\{\begin{matrix}d=-\frac{\sqrt{2}i\sqrt{\sin(x)}}{2y}\text{; }d=\frac{\sqrt{2}i\sqrt{\sin(x)}}{2y}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\text{ and }y\neq 0\\d\neq 0\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\text{ and }y=0\text{ and }x\neq 0\end{matrix}\right,$$

Resolver para x (solución compleja)

$$x=-i\ln(-i\sqrt{4\left(dy\right)^{4}-1}-2i\left(dy\right)^{2})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}\text{, }n_{1}\neq \frac{i\ln(-i\sqrt{4\left(dy\right)^{4}-1}-2i\left(dy\right)^{2})}{2\pi }$$

$$x=-i\ln(i\sqrt{4\left(dy\right)^{4}-1}-2i\left(dy\right)^{2})+2\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }d\neq 0$$

Resolver para d

$$\left\{\begin{matrix}d=\frac{\sqrt{-2\sin(x)}}{2y}\text{; }d=-\frac{\sqrt{-2\sin(x)}}{2y}\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }\left(x>2\pi n_{2}+\pi \text{ and }x<2\pi n_{2}+2\pi \right)\text{ and }y\neq 0\\d\neq 0\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\text{ and }y=0\text{ and }x\neq 0\end{matrix}\right,$$

Resolver para x

$$x=-\arcsin(2\left(dy\right)^{2})+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}\text{, }\left(n_{1}\neq 0\text{ and }|y|\leq \frac{\sqrt{2}}{2|d|}\right)\text{ or }\left(n_{1}\neq 0\text{ and }d=0\right)\text{ or }\left(d\neq 0\text{ and }y\neq 0\text{ and }|y|\leq \frac{\sqrt{2}}{2|d|}\right)$$

$$x=\arcsin(2\left(dy\right)^{2})+2\pi n_{2}+\pi \text{, }n_{2}\in \mathrm{Z}\text{, }d\neq 0\text{ and }|y|\leq \frac{\sqrt{2}}{2|d|}$$