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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(x=-i\left(y^{2}+y-1\right)^{-\frac{1}{2}}y\text{ or }x=i\left(y^{2}+y-1\right)^{-\frac{1}{2}}y\right)\text{ and }y\neq \frac{\sqrt{5}-1}{2}\text{ and }y\neq \frac{-\sqrt{5}-1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y<\frac{\sqrt{5}-1}{2}\text{ and }y>\frac{-\sqrt{5}-1}{2}\text{ and }|x|=\sqrt{-\frac{1}{y^{2}+y-1}}|y|\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-i\left(y^{2}+y-1\right)^{-\frac{1}{2}}y\text{; }x=i\left(y^{2}+y-1\right)^{-\frac{1}{2}}y\text{, }&y\neq \frac{\sqrt{5}-1}{2}\text{ and }y\neq \frac{-\sqrt{5}-1}{2}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{-\frac{1}{y^{2}+y-1}}y\text{; }x=-\sqrt{-\frac{1}{y^{2}+y-1}}y\text{, }&y>\frac{-\sqrt{5}-1}{2}\text{ and }y<\frac{\sqrt{5}-1}{2}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$