\( ( 2 x - 4 y + 6 ) d x + ( x + y - 3 ) d y = 0 \)

To solve the differential equation \[ (2x - 4y + 6) \, dx + (x + y - 3) \, dy = 0, \] we can rewrite it in the standard form: \[ M(x, y) \, dx + N(x, y) \, dy = 0, \] where \( M(x, y) = 2x - 4y + 6 \) and \( N(x, y) = x + y - 3 \). Next, we check if the equation is exact by calculating the partial derivatives: \[ \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(2x - 4y + 6) = -4, \] \[ \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(x + y - 3) = 1. \] Since \( \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} \), the equation is not exact. To make it exact, we can look for an integrating factor. A common approach is to check if the integrating factor depends only on \( x \) or \( y \). We can use the formula for the integrating factor \( \mu(y) \): \[ \mu(y) = \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N}. \] Calculating this gives: \[ \mu(y) = \frac{-4 - 1}{x + y - 3} = \frac{-5}{x + y - 3}. \] This does not depend only on \( y \), so we will try an integrating factor that depends on \( x \): \[ \mu(x) = \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M}. \] Calculating this gives: \[ \mu(x) = \frac{1 - (-4)}{2x - 4y + 6} = \frac{5}{2x - 4y + 6}. \] This also does not yield a simple integrating factor. Instead, we can try to find a solution directly by separating variables or using another method. Rearranging the original equation gives: \[ \frac{dy}{dx} = -\frac{M}{N} = -\frac{2x - 4y + 6}{x + y - 3}. \] This is a separable equation, but it may be complex to separate directly. Instead, we can try to find a solution by inspection or substitution. Let's try a substitution \( v = y - kx \) for some constant \( k \). After some trials, we can find that the solution can be expressed in terms of implicit functions or parametric forms. To summarize, the equation is not exact, and finding an integrating factor is complex. A numerical or graphical method may be more suitable for specific solutions. If you have a specific method in mind or further constraints, please let me know!