I am looking for a solution in $s$ to $$ \lambda -\frac{1}{s} +K e^t \log(\delta) \delta^s = 0 $$ Mathematica is not best pleased with this equation. If the equation were $$ 0- \frac{1}{s} +K e^t \log(\delta) \delta^s =0, $$ Mathematica tells me the solution is $s= \frac{ProductLog(\frac{1}{K e^t \log(\delta)})}{\log (\delta)}$, where ProductLog is Mathematica's implementation of the Lambert W Function, but there doesn't seem to be a closed form solution for my equation. All my variables are real; in fact $s\geq 0$ , $0<\delta<1$, $0<\lambda<1$, $t\geq 0$. $K$ is a constant of integration, will have to be chosen to satisfy another equation. A numerical approach would be almost as good as analytical; this is a first order condition to a boundary value problem. Are there any tricks I can use for convenient numerical implementation here?
For some background, the solution to this, as function of $t$, will be fed to an ODE and drive a particular value from a steady state to a boundary. So efficient numerical approaches would be highly prized.