Solve (1-x^2)/xdx=y(1+y)/(1-y)dy

$$\frac { ( 1 - x ^ { 2 } ) } { x } d x = \frac { y ( 1 + y ) } { ( 1 - y ) } d y$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&x\neq 0\text{ and }y\neq 1\\d\in \mathrm{R}\text{, }&y\neq 1\text{ and }\frac{y^{3}+y^{2}+y-1}{y-1}\geq 0\text{ and }|x|=\sqrt{\frac{y^{3}+y^{2}+y-1}{y-1}}\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(y-1\right), the least common multiple of x,1-y.

$$\left(y-1\right)\left(1-x^{2}\right)dx=-xy\left(1+y\right)dy$$

Use the distributive property to multiply y-1 by 1-x^{2}.

$$\left(y-yx^{2}-1+x^{2}\right)dx=-xy\left(1+y\right)dy$$

Use the distributive property to multiply y-yx^{2}-1+x^{2} by d.

$$\left(yd-yx^{2}d-d+x^{2}d\right)x=-xy\left(1+y\right)dy$$

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

$$ydx-ydx^{3}-dx+dx^{3}=-xy\left(1+y\right)dy$$

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

$$ydx-ydx^{3}-dx+dx^{3}=-xy^{2}\left(1+y\right)d$$

Use the distributive property to multiply -xy^{2} by 1+y.

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

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

$$ydx-ydx^{3}-dx+dx^{3}=-xy^{2}d-xy^{3}d$$

Add xy^{2}d to both sides.

$$ydx-ydx^{3}-dx+dx^{3}+xy^{2}d=-xy^{3}d$$

Add xy^{3}d to both sides.

$$ydx-ydx^{3}-dx+dx^{3}+xy^{2}d+xy^{3}d=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{\frac{y^{3}+y^{2}+y-1}{y-1}}\text{; }x=-\sqrt{\frac{y^{3}+y^{2}+y-1}{y-1}}\text{, }&y\neq \frac{\sqrt[3]{3\sqrt{33}+17}+\sqrt[3]{17-3\sqrt{33}}-1}{3}\text{ and }y\neq 1\text{ and }\frac{y^{3}+y^{2}+y-1}{y-1}\geq 0\\x\neq 0\text{, }&d=0\text{ and }y\neq 1\end{matrix}\right.$$