Why Does My Differential Equation Solution Differ from the Textbook's?

[jwxie]

Homework Statement

$\[\frac{\mathrm{dy} }{\mathrm{d} x} = \frac{x - e^{-x}}{y + e^{y}}<br /> \]$

Homework Equations

The Attempt at a Solution

This is how I did it...
[1] multiple
$\[y + e^{y} (dy) = x - e^{-x} (dx)\]$

[2] integrate both sides and I get
$\[\frac{1}{2}y^{2} + e^{y} = \frac{1}{2}x^{2} + e^{-x} +c\]$

However, the solution gives
$\[y^{2} - x^{2} +2(e^{y} - e^{-x}) = c\]$
Notice the 2? I can't figure out what I did wrong.

I thank for any helps in advance!
Thanks.

 

[Indychus]

They just solved for c, then multiplied through by 2 to eliminate the 1/2's.

 

[Dickfore]

c is an arbitrary constant, your equation is the same as their's but, with $2c \rightarrow c$. You are free to rename your arbitrary constants, as long as you do it consistently all throughout the equation.