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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&y=-x\text{ or }\left(y=-\frac{1}{x}\text{ and }x\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

$$dx+y^{2}dx+dy+x^{2}dy=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

$$dx+y^{2}dx+dy+x^{2}dy=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-y\text{, }&\text{unconditionally}\\x=-\frac{1}{y}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-y\text{, }&\text{unconditionally}\\x=-\frac{1}{y}\text{, }&y\neq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$