I am trying to solve the following differential equation (DE) analytically for two cases of $f(x)$: \begin{equation} \hspace{-35pt} y''(x) + \left( {E-{e^{-2f\left(x \right)}} - 2{e^{-f\left( x \right)}} - \delta \left( {{C_0} + {C_1}{e^{-f\left( x \right)}} + {C_2}{e^{-2f\left(x \right)}}} \right)} \right) y(x) = 0 \tag{Eq. $1$} \end{equation}
For the first case: $f(x)=x$. By making a change in variable from $x$ to $z$ by setting $z=\exp[-x]$. I transformed Eq. $1$ to the following form which I know how to solve analytically: \begin{equation} y''(z) + \frac{1}{z}y'(z) + \frac{{\left( { - A + B\,z - C\,{z^2}} \right)}}{{{z^2}}} y(z) = 0 \tag{Eq. $2$} \end{equation}
However, for the second case: $f(x)=\sinh x$, I tried the same thing by setting $z=\exp[-\sinh x]$ hoping to get a form similar to Eq. $2$, but was unable to do so. Nevertheless, for the second case, I would still like to convert it to a form similar to Eq. $2$ so that I can obtain an analytical solution. So. is there a way of doing this by using another transformation for $z$, which I had missed? Any input will be most appreciated. Thanks.