The given equation is:

$(2x - y - 3)dx + (x + 2y - 4)dy = 0$

This is a first-order linear differential equation. To determine whether it is exact, we need to check if the condition for exactness is satisfied, which is:

$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

where $M(x, y) = 2x - y - 3$ (the coefficient of $dx$) and $N(x, y) = x + 2y - 4$ (the coefficient of $dy$).

Step 1: Calculate partial derivatives

1. $\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(2x - y - 3) = -1$
2. $\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(x + 2y - 4) = 1$

Since $\frac{\partial M}{\partial y} = -1$ and $\frac{\partial N}{\partial x} = 1$, the equation is not exact.

Step 2: Approach

Since it's not exact, we need an integrating factor to make it exact. One possible method is to try an integrating factor that is a function of $x$ or $y$. Let's first try $\mu(y)$ or $\mu(x)$, which is more straightforward.

Would you like to proceed by finding an integrating factor, or explore an alternative approach for solving this differential equation?

Relative Questions:

1. How do we find an integrating factor for non-exact equations?
2. What happens if we try an integrating factor dependent on $x$?
3. How can we solve an exact differential equation after finding an integrating factor?
4. What are the conditions for a differential equation to be exact?
5. Can non-exact differential equations always be made exact with an integrating factor?

Tip: For linear differential equations, trying integrating factors based on the structure of the terms often simplifies the problem-solving process.