Thread: Solve the following ODE .. #4

1. Solve the following ODE .. #4

Problem:
Solve the following equation:

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

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

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

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

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

the equation will be:

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

Now, I stopped!

2. Multipy by $e^{-y}.$ It should be exact then.

3. Thanks.
I forgot the method of determinating the integrating factor .