Solve (1+x^2)dy+2xydx=0

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

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

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

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

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

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

$$dy+x^{2}dy+2x^{2}yd=0$$

Combine x^{2}dy and 2x^{2}yd to get 3x^{2}dy.

$$dy+3x^{2}dy=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

$$dy+x^{2}dy+2x^{2}yd=0$$

Combine x^{2}dy and 2x^{2}yd to get 3x^{2}dy.

$$dy+3x^{2}dy=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

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

Solve for x

$$x\in \mathrm{R}$$

$$y=0\text{ or }d=0$$