Differential Equations - Is the solution to $(e^y+1)^2e^{-y}dx+(e^x+1)^6e^{-x}dy=0$, $y=\ln(-5(e^x+1)^5+c)$? This is the problem. It is question 4. I typed the answer in explicit form, but my professor had said it is incorrect. Could someone check over my work, because I think I did have it correct. My Work
Because someone asked for my work due to bad handwriting:
$$(e^y+1)^2e^{-y}dx+(e^x+1)^6e^{-x}dy=0$$
Trying to seperate variables:
$$(e^y+1)^2(e^{-y})dx=-(e^x+1)^6(e^{-x})dy$$
$$\frac{-e^x}{(e^x+1)^6}dx=\frac{e^y}{(e^y+1)^2}dy$$
$$\int{\frac{-e^x}{(e^x+1)^6}dx}=\int{\frac{e^y}{(e^y+1)^2}dy}$$
Flip:
$$\int{\frac{e^y}{(e^y+1)^2}dy}=\int{\frac{-e^x}{(e^x+1)^6}dx}$$
where $u = (e^y+1), du =e^ydy$ and $v = (e^x+1), dv=e^xdx$
$$\int{u^{-2}dy}=-\int{v^{-6}}dx$$
$$-u^{-1} + c_1=\frac{1}{5}v^{-5}+c_2$$
Now, substitute:
$$-\frac{1}{e^y+1}+c_1=\frac{1}{5(e^x+1)^5}+c_2$$
The $c$ gets absorbed (also, this is the implicit solution)
$$-\frac{1}{e^y+1}=\frac{1}{5(e^x+1)^5}+c_3$$
Multiply out:
$$-5(e^x+1)^5-c_3=e^y+1$$
But $-c_3$ is the same as $+c_3$
$$-5(e^x+1)^5+c_3=e^y+1$$
$$-5(e^x+1)^5+c_3=e^y$$
Take the natural log of both sides:
$$\ln(-5(e^x+1)^5+c_3)=\ln(e^y)$$
$$\ln\left(-5(e^x+1)^5+c_3\right)=y$$
This is the answer I got. It took long to convert this to Latex, so there might be some discontinuities in the work, but the last answer is for sure what I got.

The step from:
$$ -\frac{1}{e^y + 1} = \frac{1}{5\left(e^x+1\right)^5} +c_1 \tag{1} $$
to:
$$ -5(e^x +1)^5-c_2 = e^y +1 \tag{2}$$
is wrong. Because if what you say is true then it implies that (1) and (2) are a tautology, but if you replace a term of (2) in (1):
$$ -\frac{1}{e^y + 1} = \frac{1}{5\left(e^x+1\right)^5} +c_3 $$
$$ \frac{1}{5(e^x +1)^5+c_2} = \frac{1}{5\left(e^x+1\right)^5} +c_3 $$
$$ 1 = \frac{5(e^x +1)^5+c_2}{5\left(e^x+1\right)^5} +c_3\big[5(e^x +1)^5+c_2\big] $$
$$ 1 = 1+\frac{c_2}{5\left(e^x+1\right)^5} +c_3\big[5(e^x +1)^5+c_2\big] $$
$$ 0 = \frac{c_2}{5\left(e^x+1\right)^5} +c_3\big[5(e^x +1)^5+c_2\big] $$
it is a polynomial equation if $u=e^x+1$, and the equation has a finite set of solutions of $u$ (thus $x$), and the set solution isn't a interval in $\Bbb{R}$, thus the implication from (1) to (2) is false, becaus it doesn't have sense.
