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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&ye^{y}\left(e^{x}+1\right)-xe^{x}\left(1-e^{y}\right)=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

Subtract e^{x}\left(1-e^{y}\right)dx from both sides.

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&ye^{y}\left(e^{x}+1\right)-xe^{x}\left(1-e^{y}\right)=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

Subtract e^{x}\left(1-e^{y}\right)dx from both sides.

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

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

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

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

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

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

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$