Solve 1/1-x^2*gx=x^2

$$\frac{ 1 }{ 1- { x }^{ 2 } } \times gx = { x }^{ 2 }$$

Solve for g

$$\left\{\begin{matrix}g=x-x^{3}\text{, }&|x|\neq 1\\g\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(-x-1\right).

$$1gx=\left(x-1\right)\left(-x-1\right)x^{2}$$

Use the distributive property to multiply x-1 by -x-1 and combine like terms.

$$1gx=\left(-x^{2}+1\right)x^{2}$$

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

$$1gx=-x^{4}+x^{2}$$

Reorder the terms.

$$gx=-x^{4}+x^{2}$$

The equation is in standard form.

$$xg=x^{2}-x^{4}$$

Divide both sides by x.

$$\frac{xg}{x}=\frac{x^{2}-x^{4}}{x}$$

Dividing by x undoes the multiplication by x.

$$g=\frac{x^{2}-x^{4}}{x}$$

Divide -x^{4}+x^{2} by x.

$$g=x-x^{3}$$