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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=\frac{\sqrt{8y^{2}+1}-1}{4y}\text{ or }x=-\frac{\sqrt{8y^{2}+1}+1}{4y}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=\frac{\sqrt{8y^{2}+1}-1}{4y}\text{ or }x=-\frac{\sqrt{8y^{2}+1}+1}{4y}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{8y^{2}+1}-1}{4y}\text{; }x=-\frac{\sqrt{8y^{2}+1}+1}{4y}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{8y^{2}+1}-1}{4y}\text{; }x=-\frac{\sqrt{8y^{2}+1}+1}{4y}\text{, }&y\neq 0\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$