Solve dlog(x/x)+2xsinx^2dx=0

$$d \log ( \frac { x } { x } ) + 2 x \sin x ^ { 2 } d x = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=0\text{, }&x\neq 0\\d\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\left(x=-i\sqrt{-\pi n_{1}}\text{ or }x=i\sqrt{-\pi n_{1}}\right)\text{ and }x\neq 0\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=i\sqrt{-\pi n_{1}}\text{, }n_{1}\in \mathrm{Z}\text{, }n_{1}\neq 0\text{; }x=-i\sqrt{-\pi n_{1}}\text{, }n_{1}\in \mathrm{Z}\text{, }n_{1}\neq 0\text{, }&\text{unconditionally}\\x\neq 0\text{, }&d=0\end{matrix}\right.$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&x\neq 0\\d\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }|x|=\sqrt{\pi n_{1}}\text{, }not(n_{1}<0)\text{ and }x\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-\sqrt{\pi n_{1}}\text{, }n_{1}\in \mathrm{Z}\text{, }n_{1}\geq 1\text{; }x=\sqrt{\pi n_{1}}\text{, }n_{1}\in \mathrm{Z}\text{, }n_{1}\geq 1\text{, }&\text{unconditionally}\\x\neq 0\text{, }&d=0\end{matrix}\right.$$