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

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

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&2\left(xy\right)^{2}-2yx^{3}+y^{3}+3x^{2}y^{3}-x^{3}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$