The given differential equation on the board is:

$e^{xy} \frac{dy}{dx} = e^{-y} + e^{-2x - y}$

We need to solve this using the method of separable equations. Let's break this down step by step:

Step 1: Factor the right-hand side

First, we simplify the right-hand side:

$e^{-y} + e^{-2x - y} = e^{-y}(1 + e^{-2x})$

Thus, the equation becomes:

$e^{xy} \frac{dy}{dx} = e^{-y}(1 + e^{-2x})$

Step 2: Isolate $\frac{dy}{dx}$

Divide both sides by $e^{xy}$ to isolate $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{e^{-y}(1 + e^{-2x})}{e^{xy}} = e^{-y - xy}(1 + e^{-2x})$

Simplifying the exponent on the right-hand side:

$\frac{dy}{dx} = e^{-y(1 + x)}(1 + e^{-2x})$

Step 3: Separate variables

To separate the variables, we aim to have all $y$-related terms on one side and all $x$-related terms on the other. This gives:

$\frac{dy}{e^{-y(1 + x)}} = (1 + e^{-2x}) dx$

Rewriting the left-hand side:

$e^{y(1 + x)} dy = (1 + e^{-2x}) dx$

Step 4: Integrate both sides

Now, integrate both sides. For the left-hand side, we integrate with respect to $y$, and for the right-hand side, we integrate with respect to $x$:

$\int e^{y(1 + x)} dy = \int (1 + e^{-2x}) dx$

The solution to the integrals will give us the general solution to the differential equation.

Would you like me to proceed with calculating the integrals, or would you like further clarification on the steps?