I have this equation: $$ p(y) = -\left(e^{-\tfrac{K_{py}zy}{p_u+c e}}-1\right)\left(p_u+c e^1\right)-c\left(1-e^{-y}\right)^d\left(e^1-e^{1-y}\right) $$ The two unknowns are $c$ and $d$ and the system that I want to solve is: \begin{cases} p(y_m) = p_m\\ \\ \dfrac{\mathrm{d}p(y)}{\mathrm{d}y}\Bigg|_{y = y_m} = 0 \end{cases}
I already seen what you suggest in this question but again I don't understand how to solve the problem even with Lambert's W function. Considering that all the quantities are real constant and the following variables are $y\geq0$, $y_m>0$, $p_u\geq0$, $p_m\geq p_u$, does exist an exact solution? Can you help me to find it?
Thank you for your kind cooperation.