Hi,

I am trying to solve the following differential equation:

$\displaystyle \frac {dy}{dx} = e^{x+y}$

Now:

$\displaystyle \frac {dy}{dx} = e^x e^y$

$\displaystyle \frac {1}{e^y} dy = e^x dx$

$\displaystyle \int \frac {1}{e^y} dy = \int e^x dx$

$\displaystyle -e^{-y} = e^x + C$

$\displaystyle ln(e^{-y^{-1})} = ln(e^x + C)$

$\displaystyle \frac{-1}{y} = x + ln(C)$

$\displaystyle y = \frac {1}{-x - ln(C)}$

But the answer in the book is shown as:

$\displaystyle e^{x+y}+Ce^y+1=0$

Where am I going wrong?