$$2 x ( y e ^ { x ^ { 2 } } - 1 ) d x + e ^ { 2 } d y = 0$$
$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y=\frac{2x^{2}}{2x^{2}e^{x^{2}}+e^{2}}\end{matrix}\right.$
$$2x^{2}\left(ye^{x^{2}}-1\right)d+e^{2}dy=0$$
$$\left(2x^{2}ye^{x^{2}}-2x^{2}\right)d+e^{2}dy=0$$
$$2x^{2}ye^{x^{2}}d-2x^{2}d+e^{2}dy=0$$
$$\left(2x^{2}ye^{x^{2}}-2x^{2}+e^{2}y\right)d=0$$
$$\left(e^{2}y+2yx^{2}e^{x^{2}}-2x^{2}\right)d=0$$
$$d=0$$
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