Let's solve the differential equation:

$(e^{2y} - y) \cdot \cos(x) \cdot \frac{dy}{dx} = e^y \cdot \sin(2x)$

Step 1: Separate variables

The equation involves both $y$ and $x$ in a non-trivial way, but we will try to isolate the terms involving $y$ on one side and the terms involving $x$ on the other side. First, divide both sides of the equation by $\cos(x)$:

$(e^{2y} - y) \cdot \frac{dy}{dx} = \frac{e^y \cdot \sin(2x)}{\cos(x)}$

Since $\frac{\sin(2x)}{\cos(x)} = 2 \cdot \tan(x)$, we can simplify the right-hand side:

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

Step 2: Separate the variables

Now we need to isolate the terms involving $y$ and those involving $x$. Divide both sides by $e^{2y} - y$:

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

At this point, the equation is separable. We can rearrange the terms:

$\frac{e^{2y} - y}{e^y \cdot \tan(x)} \, dy = 2 dx$

Step 3: Integrate both sides

Now we need to integrate both sides of the equation. We have:

$\int \frac{e^{2y} - y}{e^y \cdot \tan(x)} \, dy = \int 2 \, dx$

Step 4: Solve the integrals

We will now need to solve both integrals. However, the integrand involving $y$ is quite complicated. Therefore, this solution will require more advanced techniques or numerical methods to evaluate. If you're looking for further detail in solving the integral, let me know and we can attempt it more specifically.

Final Answer:

We have reached the form:

$\int \frac{e^{2y} - y}{e^y \cdot \tan(x)} \, dy = 2x + C$

Where $C$ is the constant of integration.

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Would you like me to help with further details on solving the integrals, or would you prefer another explanation? Here's a set of questions for further exploration:

1. How do we simplify the integral involving $y$?
2. What is the relationship between the functions $e^y$, $\tan(x)$, and $y$ in such equations?
3. Can we apply a substitution method for easier integration in this type of equation?
4. How do separation of variables typically work in more straightforward differential equations?
5. What methods can we use to numerically approximate solutions for complex integrals like this one?

Tip: For equations involving both exponential and trigonometric functions, a substitution or series expansion might be a useful strategy.