I have a differential equation with variables x(t) and y(t), and a following differential equation:
$$xy \frac{d}{dt}\Big{(}\frac{x}{y}\Big{)} = c_{1} (y-x)-c_{2} \quad c_{1}~ \text{and}~c_{2}~\text{are}~\text{constants}$$
This equation comes from a coupled differential equation system, which describes the derivatives of x(t) and y(t).
Maybe a new variable which contains x(t) and y(t) can help to find the solution of this equation instead of solving the original equation system.
(The original system is $$\frac{dx}{dt} = \gamma xy-\alpha xy-\gamma x^{2}-\gamma\beta x\qquad(1)\\ \frac{dy}{dx} = \gamma xy-\alpha xy-\gamma y^{2}\qquad (2)$$
I tried to simplifying the system with these steps: $$ \frac{\dot{x}}{x} = \gamma y-\alpha y-\gamma x-\gamma\beta \qquad(1) \\ \frac{\dot{y}}{y} = \gamma x-\alpha x-\gamma y \qquad(2)\\ (1)-(2) \rightarrow \frac{\dot{x}}{x}-\frac{\dot{y}}{y} = (2\gamma-\alpha)(y-x)-\gamma\beta\\ \frac{\dot{x}y-\dot{y}x}{xy} = (2\gamma-\alpha)(y-x)-\gamma\beta\\ \frac{y}{x}\frac{d}{dt}\Big(\frac{x}{y}\Big) = (2\gamma-\alpha)(y-x)-\gamma\beta \\ \frac{y}{x}\frac{d}{dt}\Big{(}\frac{x}{y}\Big{)} = c_{1} (y-x)-c_{2} \qquad \text{where}~c_{1} =2\gamma-\alpha\qquad~c_{2} = \gamma\beta $$