Consider the equation $$\delta e^{\delta x}+ \frac{(1-e^{\delta x})(-ae^{-ax}+(a+\delta)e^{(a+\delta)x} +\delta e^{\delta x})}{(1+e^{-ax}+e^{(a+\delta)x}+e^{\delta x})} =0$$
Denoting $g(x)=1-e^{\delta x}$ and $h(x)=(1+e^{-ax}+e^{(a+\delta)x}+e^{\delta x})$, we can rewrite the equation:
$$-g'+g\frac{h'}{h}=0$$
or more conveniently:
$$gh'-g'h=0$$
We know $g(0)=0, h(0)=4$
I thought of maybe using the Laplace transform but then I recalled that I must use convolutions to transform a product. I tried solving without using $g,h$, and got monstrous expressions.
Any ideas on how to solve it? It's been more than 10 years since my differential equations course :(
(this is a follow-up question to my original one)