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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(x=-i\sqrt{y}e^{\frac{y}{2}}\left(ye^{y}-2e^{y}-2\right)^{-\frac{1}{2}}\text{ or }x=i\sqrt{y}e^{\frac{y}{2}}\left(ye^{y}-2e^{y}-2\right)^{-\frac{1}{2}}\right)\text{ and }ye^{y}-2e^{y}-2\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-i\sqrt{y}e^{\frac{y}{2}}\left(ye^{y}-2e^{y}-2\right)^{-\frac{1}{2}}\text{; }x=i\sqrt{y}e^{\frac{y}{2}}\left(ye^{y}-2e^{y}-2\right)^{-\frac{1}{2}}\text{, }&ye^{y}-2e^{y}-2\neq 0\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=e^{\frac{y}{2}}\sqrt{-\frac{y}{ye^{y}-2e^{y}-2}}\text{; }x=-e^{\frac{y}{2}}\sqrt{-\frac{y}{ye^{y}-2e^{y}-2}}\text{, }&\left(y-2\right)e^{y}\neq 2\text{ and }\frac{ye^{y}}{ye^{y}-2e^{y}-2}\leq 0\text{ and }ye^{y}-2e^{y}-2\neq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$