Solve (x-1)^2ydx+x^2(y+1)dy=0

$$( x - 1 ) ^ { 2 } y d x + x ^ { 2 } ( y + 1 ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=\frac{\sqrt{\left(y-1\right)^{2}-4}-y+1}{2}\text{ or }x=\frac{-\sqrt{\left(y-1\right)^{2}-4}-y+1}{2}\text{ or }x=0\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}-2x+1\right)ydx+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2}-2x+1 by y.

$$\left(x^{2}y-2xy+y\right)dx+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2}y-2xy+y by d.

$$\left(x^{2}yd-2xyd+yd\right)x+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2}yd-2xyd+yd by x.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2} by y+1.

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

Use the distributive property to multiply x^{2}y+x^{2} by d.

$$ydx^{3}-2ydx^{2}+ydx+\left(x^{2}yd+x^{2}d\right)y=0$$

Use the distributive property to multiply x^{2}yd+x^{2}d by y.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}dy^{2}+x^{2}dy=0$$

Combine -2ydx^{2} and x^{2}dy to get -ydx^{2}.

$$ydx^{3}-ydx^{2}+ydx+x^{2}dy^{2}=0$$

Combine all terms containing d.

$$\left(yx^{3}-yx^{2}+yx+x^{2}y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{2}+xy+yx^{3}-yx^{2}\right)d=0$$

Divide 0 by yx^{3}-yx^{2}+yx+x^{2}y^{2}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=\frac{\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\geq 3\right)\text{ or }\left(x=\frac{\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\leq -1\right)\text{ or }\left(x=\frac{-\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\geq 3\right)\text{ or }\left(x=\frac{-\sqrt{y^{2}-2y-3}-y+1}{2}\text{ and }y\leq -1\right)\text{ or }x=0\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}-2x+1\right)ydx+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2}-2x+1 by y.

$$\left(x^{2}y-2xy+y\right)dx+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2}y-2xy+y by d.

$$\left(x^{2}yd-2xyd+yd\right)x+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2}yd-2xyd+yd by x.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}\left(y+1\right)dy=0$$

Use the distributive property to multiply x^{2} by y+1.

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

Use the distributive property to multiply x^{2}y+x^{2} by d.

$$ydx^{3}-2ydx^{2}+ydx+\left(x^{2}yd+x^{2}d\right)y=0$$

Use the distributive property to multiply x^{2}yd+x^{2}d by y.

$$ydx^{3}-2ydx^{2}+ydx+x^{2}dy^{2}+x^{2}dy=0$$

Combine -2ydx^{2} and x^{2}dy to get -ydx^{2}.

$$ydx^{3}-ydx^{2}+ydx+x^{2}dy^{2}=0$$

Combine all terms containing d.

$$\left(yx^{3}-yx^{2}+yx+x^{2}y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{2}+xy+yx^{3}-yx^{2}\right)d=0$$

Divide 0 by yx^{3}-yx^{2}+yx+x^{2}y^{2}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{-\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{; }x=0\text{; }x=\frac{\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{; }x=\frac{-\sqrt{\left(y-3\right)\left(y+1\right)}-y+1}{2}\text{, }&y\leq -1\text{ or }y\geq 3\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$