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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-\sqrt{3}y\text{ or }x=\sqrt{3}y\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

$$dx^{3}-y^{2}dx-2xy^{2}d=0$$

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

$$dx^{3}-3y^{2}dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&|x|=\sqrt{3}|y|\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

$$dx^{3}-y^{2}dx-2xy^{2}d=0$$

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

$$dx^{3}-3y^{2}dx=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\sqrt{3}y\text{; }x=0\text{; }x=-\sqrt{3}y\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-\sqrt{3}y\text{; }x=0\text{; }x=\sqrt{3}y\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$