19. dy/dx = (xy + 3x - y - 3)/(xy - 2x + 4y - 8)

To solve this differential equation, we can start by factoring or simplifying the given expression. First, note the numerator \(x y + 3 x - y - 3\) and the denominator \(x y - 2 x + 4 y - 8\). Let's make the substitution \(y = vx\), where \(v\) is a function of \(x\). Then \(dy/dx = v + x(dv/dx)\). This transforms the equation into a separable form, allowing us to solve for \(v\) in terms of \(x\). After substituting and simplifying, we can separate the variables and integrate both sides. The constants of integration will give rise to the general solution for \(y\) in terms of \(x\). In real-world applications, understanding how to approach such a first-order differential equation can model various phenomena, from population dynamics to physics problems like motion under gravity where variables interact in complex ways.