To solve the given differential equation using the Separation of Variables method, let's break it down:

The equation is:

$\frac{dy}{dx} + 2xy = x, \quad y(0) = 2$

Step 1: Rearrange the equation

Rewrite the equation to isolate $\frac{dy}{dx}$:

$\frac{dy}{dx} = x - 2xy$

Factor out $y$ on the right-hand side:

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

This equation cannot directly be solved using the separation method. Would you like ShareUpon review, it seems this equation cannot be directly solved using the Separation of Variables method, as it's not separable in its current form. It is more suited for an Integrating Factor Method.

Would you like me to proceed with the Integrating Factor Method or another approach (e.g., numerical methods like Euler's method)? Let me know your preference!

Alternatively, feel free to clarify any part of the question or request further steps for solving the problem.