Solve y(x^2+xy-2y^2)dx+x(3y^2-xy-x^2)dy=0

$$y ( x ^ { 2 } + x y - 2 y ^ { 2 } ) d x + x ( 3 y ^ { 2 } - x y - x ^ { 2 } ) d y = 0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply yx^{2}+xy^{2}-2y^{3} by d.

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

Use the distributive property to multiply yx^{2}d+xy^{2}d-2y^{3}d by x.

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

Use the distributive property to multiply x by 3y^{2}-xy-x^{2}.

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

Use the distributive property to multiply 3xy^{2}-yx^{2}-x^{3} by d.

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

Use the distributive property to multiply 3xy^{2}d-yx^{2}d-x^{3}d by y.

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

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

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

Combine y^{2}dx^{2} and -x^{2}dy^{2} to get 0.

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

Combine ydx^{3} and -x^{3}dy to get 0.

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

The equation is in standard form.

$$xy^{3}d=0$$

Divide 0 by y^{3}x.

$$d=0$$

Solve for x (complex solution)

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

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply yx^{2}+xy^{2}-2y^{3} by d.

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

Use the distributive property to multiply yx^{2}d+xy^{2}d-2y^{3}d by x.

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

Use the distributive property to multiply x by 3y^{2}-xy-x^{2}.

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

Use the distributive property to multiply 3xy^{2}-yx^{2}-x^{3} by d.

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

Use the distributive property to multiply 3xy^{2}d-yx^{2}d-x^{3}d by y.

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

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

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

Combine y^{2}dx^{2} and -x^{2}dy^{2} to get 0.

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

Combine ydx^{3} and -x^{3}dy to get 0.

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

The equation is in standard form.

$$dy^{3}x=0$$

Divide 0 by y^{3}d.

$$x=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply yx^{2}+xy^{2}-2y^{3} by d.

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

Use the distributive property to multiply yx^{2}d+xy^{2}d-2y^{3}d by x.

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

Use the distributive property to multiply x by 3y^{2}-xy-x^{2}.

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

Use the distributive property to multiply 3xy^{2}-yx^{2}-x^{3} by d.

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

Use the distributive property to multiply 3xy^{2}d-yx^{2}d-x^{3}d by y.

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

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

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

Combine y^{2}dx^{2} and -x^{2}dy^{2} to get 0.

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

Combine ydx^{3} and -x^{3}dy to get 0.

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

The equation is in standard form.

$$xy^{3}d=0$$

Divide 0 by y^{3}x.

$$d=0$$

Solve for x

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

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply yx^{2}+xy^{2}-2y^{3} by d.

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

Use the distributive property to multiply yx^{2}d+xy^{2}d-2y^{3}d by x.

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

Use the distributive property to multiply x by 3y^{2}-xy-x^{2}.

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

Use the distributive property to multiply 3xy^{2}-yx^{2}-x^{3} by d.

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

Use the distributive property to multiply 3xy^{2}d-yx^{2}d-x^{3}d by y.

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

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

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

Combine y^{2}dx^{2} and -x^{2}dy^{2} to get 0.

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

Combine ydx^{3} and -x^{3}dy to get 0.

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

The equation is in standard form.

$$dy^{3}x=0$$

Divide 0 by y^{3}d.

$$x=0$$