Solve Separable Differential Equation (e^y + 1)^2e^(-y)dx + (e^x + 1)^2e^(-x)dy = 0

To solve this differential equation using the separable method, we need to write it in the form of where and are functions of only and , respectively. Starting with the given equation:

We can divide both sides by to get:

Now we can integrate both sides with respect to their respective variables:

The integral on the left can be evaluated using substitution:

Let , then and the integral becomes:

Similarly, the integral on the right can be evaluated using substitution:

Let , then and the integral becomes:

Substituting these back into the original equation, we get:

where .

Therefore, the solution to the differential equation is:

where is the constant of integration.