My question is how we can find the analytical solution of the following system $$ c \ln\left(\dfrac{u}{y}\right)= J x,$$ $$c(u-y)= \left(\dfrac{A}{B x+ D} x + G\right) x,$$ where $u,c, D, A$ et $J$ are constants, and $x$ et $y$ are uknowns. I try to use Wolfram Alpha but it answer me to give a real values of parameters
We are note interested by the trivial solution $u=y$ and $x=0$
Thank you in advance for the help
Let $y=\frac u z$. So, the first equation gives $z=e^{\frac{J x}{c}}$.
Plug in the second equation and isolate the exponential term $$e^{-\frac{J x}{c}}=\frac {-(A+B G)x^2+ (B c u-D G)x+c d u } {cu(Bx+D) }$$
the solution of which being given in terms of the generalized Lambert function (have a look at equation $(4)$ in the paper).