Question

I need to differentiate y'=(x-e^-x)/(y+e^y)

Answer 1

Differentiate? So its already dy/dx and you need d^2y/dx^2?

Answer 2

the question is solve the differential equation

Answer 3

Okay, then not at all derivative and such. The goal for this problem is to gather all the y expressions with dy and all the x expressions with dx and then integrate once you do so..
So first thing first, y' is the same as dy/dx. Now all we need to do is get all the y's with y's and the x's with x's. In the end, this is called separation of variables. Well, this can easily be done. If I multiply both sides by y+e^y I get:

\[\frac{ dy(y+e^{y}) }{ dx }=x-e^{-x}\]Now all I need to do is multiply both sides by dx
\[dy(y+e^{y}) = dx(x-e^{-x})\]Now that weve grouped everything, we just integrate both sides:

\[\int\limits_{}^{}y + \int\limits_{}^{}e^{y}=\int\limits_{}^{}x - \int\limits_{}^{}e^{-x}\]Which becomes:
\[\frac{ y^{2} }{ 2 }+e^{y}= \frac{ x^{2} }{ 2 }+e^{-x} + C\]You cant really do too much to isolate y, so that should be all the work you need to do :3