While trying to obtain an analytical solution for a system of 1st order ODEs from this question, I have gained an extra constant (really, two of them), which I don't know how to find.
The original systems is:
$$\dot{x} = Ax - Bxy \\ \dot{y} = Cxy - Dy$$
To separate the functions, I differentiate the equation for $y$ and use several identities to obtain:
$$\ddot{y}=\frac{\dot{y}^2}{y}+(A-By)(\dot{y}+Dy)$$
Which I then integrate and get an implicit 1st order equation:
$$\frac{\dot{y}}{y}-D \ln \left(\frac{\dot{y}}{y}+D \right)=A \ln y-B y+c_1$$
Using Lambert function, I can solve for the derivative:
$$\dot{y}=- Dy \left(1+W \left(-\frac{e^{-1-c_1/D}}{D} y^{-A/D} \exp \left(\frac{B}{D} y \right) \right) \right)$$
Which formally offers a full analytical solution $t(y)$ as an indefinite integral (even though it's not likely to have an exact form in terms of known functions).
How do I find the constant $c_1$? What 'initial conditions' should I use?
Because of the symmetry, we can repeat the same procedure for $x$ changing $C \leftrightarrow -B$ and $D \leftrightarrow -A$.
$$\frac{\dot{x}}{x}+A \ln \left(\frac{\dot{x}}{x}-A \right)=-D \ln x+C x+c_2$$
So, I suppose, there should be a way to find both $c_1$ and $c_2$ from the original system. But I wasn't able to do that yet.
I'm also aware that there are some problems with the results I obtained as the logarithms (and consequently Lambert function) will not always be defined in real numbers.