Solve 6xdx-6ydy=3x^2ydy-2xy^2dx

$$6 x d x - 6 y d y = 3 x ^ { 2 } y d y - 2 x y ^ { 2 } d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(x=-6\left(36-6y^{2}\right)^{-\frac{1}{2}}y\text{ or }x=6\left(36-6y^{2}\right)^{-\frac{1}{2}}y\right)\text{ and }y\neq -\sqrt{6}\text{ and }y\neq \sqrt{6}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$6x^{2}d-6ydy=3x^{2}ydy-2xy^{2}dx$$

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

$$6x^{2}d-6y^{2}d=3x^{2}ydy-2xy^{2}dx$$

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

$$6x^{2}d-6y^{2}d=3x^{2}y^{2}d-2xy^{2}dx$$

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

$$6x^{2}d-6y^{2}d=3x^{2}y^{2}d-2x^{2}y^{2}d$$

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

$$6x^{2}d-6y^{2}d=x^{2}y^{2}d$$

Subtract x^{2}y^{2}d from both sides.

$$6x^{2}d-6y^{2}d-x^{2}y^{2}d=0$$

Reorder the terms.

$$-dx^{2}y^{2}+6dx^{2}-6dy^{2}=0$$

Combine all terms containing d.

$$\left(-x^{2}y^{2}+6x^{2}-6y^{2}\right)d=0$$

Divide 0 by 6x^{2}-6y^{2}-x^{2}y^{2}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y>-\sqrt{6}\text{ and }|y|<\sqrt{6}\text{ and }|x|=\frac{6|y|}{\sqrt{36-6y^{2}}}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$6x^{2}d-6ydy=3x^{2}ydy-2xy^{2}dx$$

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

$$6x^{2}d-6y^{2}d=3x^{2}ydy-2xy^{2}dx$$

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

$$6x^{2}d-6y^{2}d=3x^{2}y^{2}d-2xy^{2}dx$$

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

$$6x^{2}d-6y^{2}d=3x^{2}y^{2}d-2x^{2}y^{2}d$$

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

$$6x^{2}d-6y^{2}d=x^{2}y^{2}d$$

Subtract x^{2}y^{2}d from both sides.

$$6x^{2}d-6y^{2}d-x^{2}y^{2}d=0$$

Reorder the terms.

$$-dx^{2}y^{2}+6dx^{2}-6dy^{2}=0$$

Combine all terms containing d.

$$\left(-x^{2}y^{2}+6x^{2}-6y^{2}\right)d=0$$

Divide 0 by 6x^{2}-6y^{2}-x^{2}y^{2}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\sqrt{6}\left(6-y^{2}\right)^{-\frac{1}{2}}y\text{; }x=\sqrt{6}\left(6-y^{2}\right)^{-\frac{1}{2}}y\text{, }&y\neq -\sqrt{6}\text{ and }y\neq \sqrt{6}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{\frac{6}{6-y^{2}}}y\text{; }x=-\sqrt{\frac{6}{6-y^{2}}}y\text{, }&|y|<\sqrt{6}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$