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The given differential equation is:

\[
(2x - y) \, dx + (2y - 3x) \, dy = 0
\]

This is a first-order linear differential equation in terms of both \(x\) and \(y\). To solve it, we will check if it's exact, and if not, we can try to solve it using integrating factors or other methods.

### Step 1: Check for exactness
A differential equation of the form \( M(x,y) \, dx + N(x,y) \, dy = 0 \) is exact if:

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]

Here, 
- \( M(x, y) = 2x - y \)
- \( N(x, y) = 2y - 3x \)

Now, we compute the partial derivatives:
\[
\frac{\partial M}{\partial y} = -1
\]
\[
\frac{\partial N}{\partial x} = -3
\]

Since \(\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}\), the equation is **not exact**.

### Step 2: Try a suitable substitution or integrating factor
We might need to find an integrating factor to solve this non-exact equation. One possible approach is to look for an integrating factor that depends only on \(x\) or \(y\), or test other methods.

Would you like to proceed with exploring integrating factors, or do you want more details on any specific step so far?

### Related Questions:
1. What does it mean for a differential equation to be exact?
2. How do you find an integrating factor for a non-exact equation?
3. Can substitution techniques simplify certain types of differential equations?
4. What is the general solution of a first-order linear differential equation?
5. How do you verify the correctness of a solution to a differential equation?

**Tip:** When solving non-exact differential equations, finding an appropriate integrating factor can often transform the problem into an exact one, making it easier to solve.

Learn how to solve the non-exact differential equation (2x - y)dx + (2y - 3x)dy = 0. Explore methods such as exactness check and integrating factors to find the solution. Ideal for undergraduate students studying differential equations.

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