The equation on the whiteboard is:

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

This is a first-order differential equation, and to solve it, we can try separating variables or look for an integrating factor. Here's a suggested approach to solve this:

Step 1: Rearranging the equation

We can isolate $\frac{dy}{dx}$ on one side:

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

Now, let's simplify the right-hand side:

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

Simplifying each term:

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

At this point, we could either attempt variable separation or use another method to simplify the integration process.

Would you like a step-by-step continuation with this approach?

Here are 5 related questions that could further enhance your understanding:

1. What is the general method for solving separable differential equations?
2. How do you know if an equation can be separated in variables?
3. What is an integrating factor, and how does it help in solving differential equations?
4. How do we solve non-linear first-order differential equations?
5. What role does substitution play in simplifying complex differential equations?

Tip: When solving differential equations, always check whether the equation is exact, separable, or can be simplified by substitution. This will guide you toward the best solving technique.