Solve 2xydy-(x^2-y^2+1)dx=0

$$2 x y d y - ( x ^ { 2 } - y ^ { 2 } + 1 ) d x = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-\sqrt{3y^{2}-1}\text{ or }x=\sqrt{3y^{2}-1}\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$2xy^{2}d-\left(x^{2}-y^{2}+1\right)dx=0$$

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

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

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

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

To find the opposite of dx^{3}-y^{2}dx+dx, find the opposite of each term.

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

Combine 2xy^{2}d and y^{2}dx to get 3xy^{2}d.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

$$2xy^{2}d-\left(x^{2}-y^{2}+1\right)dx=0$$

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

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

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

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

To find the opposite of dx^{3}-y^{2}dx+dx, find the opposite of each term.

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

Combine 2xy^{2}d and y^{2}dx to get 3xy^{2}d.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\sqrt{3y^{2}-1}\text{; }x=0\text{; }x=-\sqrt{3y^{2}-1}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

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