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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

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

Multiply y and y to get y^{2}.

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

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

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

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

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

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

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

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

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

Combine x^{2}y^{2}d and y^{2}x^{2}d to get 2x^{2}y^{2}d.

$$2x^{2}y^{2}d-x^{2}d-y^{2}d=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Multiply x and x to get x^{2}.

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

Multiply y and y to get y^{2}.

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

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

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

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

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

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

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

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

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

Combine x^{2}y^{2}d and y^{2}x^{2}d to get 2x^{2}y^{2}d.

$$2x^{2}y^{2}d-x^{2}d-y^{2}d=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for x (complex solution)

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

Solve for x

$$\left\{\begin{matrix}x=\frac{y}{\sqrt{2y^{2}-1}}\text{; }x=-\frac{y}{\sqrt{2y^{2}-1}}\text{, }&|y|>\frac{\sqrt{2}}{2}\\x=0\text{, }&y=0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$