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.