To solve the differential equation:

\[
(x + 2y) \, dx = x \, dy
\]

we can rearrange and simplify it to separate variables. Here's how:

1. **Rewrite the equation**:  
   Divide both sides by \( x \, dx \) (assuming \( x \neq 0 \)):
   \[
   \frac{x + 2y}{x} \, dx = dy
   \]

2. **Simplify the left side**:  
   This gives:
   \[
   \left(1 + \frac{2y}{x}\right) dx = dy
   \]

3. **Separate variables**:
   Rearrange terms to isolate \( y \) on one side and \( x \) on the other:
   \[
   \frac{dy}{dx} = 1 + \frac{2y}{x}
   \]

4. **Solve the equation using an appropriate method**:  
   This is a first-order linear differential equation in \( y \) and \( x \). We can solve it using an integrating factor, substitution, or recognizing a simpler solution path based on its form.

Would you like a detailed solution of this using an integrating factor or substitution? 

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Here are 5 related questions for further exploration:

1. What is an integrating factor, and how does it help solve differential equations?
2. How would you solve this equation if \( y \) were a constant?
3. Can this equation be solved by separation of variables directly?
4. What if we added a constant term to \( (x + 2y)dx = xdy \)? How would that change the solution approach?
5. How does the substitution \( u = \frac{y}{x} \) simplify certain differential equations?

**Tip:** In a differential equation, isolating terms with \( dx \) and \( dy \) often reveals methods like separation of variables or allows for substitutions that simplify the equation

Explore how to solve the differential equation (x + 2y)dx = xdy through methods like separation of variables and integrating factors. This detailed step-by-step solution covers essential concepts in differential equations, suitable for undergraduate students.

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