Solve x^2ydx=(x^2-y^2-xy^2+1)dy

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

Solve for d

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

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{3}yd=\left(x^{2}-y^{2}-xy^{2}+1\right)dy$$

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

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

Use the distributive property to multiply \left(x^{2}-y^{2}-xy^{2}\right)d+d by y.

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

Subtract \left(x^{2}-y^{2}-xy^{2}\right)dy from both sides.

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

Subtract dy from both sides.

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

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

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

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

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

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

$$x^{3}yd-x^{2}dy+dy^{3}+xdy^{3}-dy=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$