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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

Combine -y^{2}dx and -2xdy^{2} to get -3y^{2}dx.

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=\sqrt{2}y-y\text{ or }x=-\sqrt{2}y-y\text{ or }x=y\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

Combine -y^{2}dx and -2xdy^{2} to get -3y^{2}dx.

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\sqrt{2y^{2}}-y\text{; }x=y\text{; }x=-\sqrt{2y^{2}}-y\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

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