In this solution for an exact differential equation I can't reconcile my answer with the book's answer. Please help.
I've never seen the solution method that you have presented so I can't find if there's an error in it. On the other hand if we look at the solution method I know I do not get your book answer either. The fix is trivial and I suspect a typo. I'll run through how I know how to do it and maybe you can compare the two methods to find where your problem is.
I won't bother to prove that your original equation is exact. I'm assuming you already checked that. So I will start with the assumption that there is a function F(x, y) such that
$\displaystyle \frac{\partial F}{\partial x} = P(x, y) = e^x sin(y) + e^{-y}$
which means that
$\displaystyle F(x, y) = \int P(x, y) \partial x + \phi (y)$
(The $\displaystyle \partial x$ in the integrand is merely to specify we are keeping the value of y fixed during the integration.)
This leads to
$\displaystyle F(x, y) = e^x sin(y) + xe^{-y} + \phi (y)$
We also know that
$\displaystyle \frac{\partial F}{\partial y} = Q(x, y)$
Integrating this out we get that $\displaystyle \phi (y) = C$
So the final solution will be
$\displaystyle F(x, y) = e^x sin(y) + xe^{-y} + C$
whereas the book claims $\displaystyle F(x, y) = C = e^x sin(y) + xe^{-y}$. Hence the suspicion of a typo.
-Dan