The following is an equation I've derived in my personal research:
$$ \frac{d^2V}{dx^2}=e^{\alpha x} \sinh(V) $$
I'm looking to solve it explicitly for V(x). It's a variant of the Poisson-Boltzmann equation, which has the form $ \frac{d^2V}{dx^2}= \sinh(V) $, and can be solved by multiplying both sides by $ \frac{dV}{dx} $ and integrating, rearranging and integrating again.
I've tried several methods, changes of variables, separation of variables, integration by parts, etc. The best I can do is a change of variable, $ y=e^{\alpha x} $ which yields:
$$ \alpha^2(V_{yy} \cdot y +V_v) = \sinh(V) $$
OR equivalently...
$$ \alpha^2 \frac{d}{dy}(yV_y) = \sinh(V) $$
But again I can't get any further than this.
A) Can it be solved explicitly? If so, how?
B) Is there a name for this equation so I can find other information on it and it's properties?
I have a suspicion that some change to a complex variable and contour integrals might be involved in solving this but I can't be sure since I took a took an introductory complex analysis class almost 10 years ago.