:Problem

Solve the following equation:

$\displaystyle (1+e^y \cdot e^{\left( e^x \right)}) dx - dy = 0$

:Solution

I multiply the equation by $\displaystyle e^x$, to get:

$\displaystyle (e^x+e^y \, e^x \cdot e^{\left( e^x \right)}) dx - e^x dy = 0$

$\displaystyle e^xdx + e^y e^x e^{\left( e^x \right)} dx - e^x dy = 0$

Let $\displaystyle t=e^x \implies dt=e^xdx$

the equation will be:

$\displaystyle dt+e^ye^tdt-tdy=0$

Now, I stopped!