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

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

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

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

To find the opposite of y^{2}d+y^{2}x^{2}d, find the opposite of each term.

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

Combine x^{2}y^{2}d and -y^{2}x^{2}d to get 0.

$$x^{2}d-y^{2}d=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&|x|=|y|\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

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

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

To find the opposite of y^{2}d+y^{2}x^{2}d, find the opposite of each term.

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

Combine x^{2}y^{2}d and -y^{2}x^{2}d to get 0.

$$x^{2}d-y^{2}d=0$$

Combine all terms containing d.

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

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

$$d=0$$

Solve for x (complex solution)

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

Solve for x

$$\left\{\begin{matrix}\\x=-y\text{; }x=y\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$