$y - y_0 = \frac{d^2}{dx^2}\left[ \ln\left( \frac{y}{y_0} \right) \right]$, solve for $y$. I'm looking at a simple differential equation of the form:
$$y - y_0 = \frac{d^2}{dx^2}\left[ \ln\left( \frac{y}{y_0} \right) \right].\tag{$\star$}$$
Here $y=y(x)$ is a function of $x$ only, $y_0$ is a constant, and $x,y,y_0 \in \mathbb{R}$ are real.
After an hour or so working on a problem, Eq. $\left(\star\right)$ popped out. I haven't made a significant effort to solve this DE just yet, because it is so simple looking and I thought it'd be a nice question for this site. This is "work-work", not "homework".
Is there a name for this DE? Are there simple solutions, besides the trivial solution $y(x) = y_0$?

Maple yields this implicit solution:
$$
\int ^{y \left( x \right) }\!{\frac {1}{t\sqrt {2\,t-2\,y_{{0}}\ln 
 \left( t \right) +B}}\;}{dt}=x+C
$$
where $B,C$ are constants.
