Another nonlinear ODE of second order.

Another nonlinear ODE of second order.

I have problems finding out whether this initial value problem has an
explicit form solution or if it is possible to grind out a term-by-term
representation of this solution using power series expansions.
\begin{equation}f^{\prime\prime}(x)-\frac{f(x)-a}{b}f^{\prime}(x)^2=0,\quad
x\in(0,1),\qquad f(1/2)=a,\,f^{\prime}(1/2)=\sqrt{2\pi b}.\end{equation}
where $a\in\mathbb{R}$ and $b>0$ are constants.
I have tried the following: Find a solution for the easier problem,
\begin{equation}f^{\prime\prime}(x)-\frac{f(x)-a}{b}f^{\prime}(x)=0,\quad
x\in(0,1),\qquad f(1/2)=a,\,f^{\prime}(1/2)=\sqrt{2\pi b}.\end{equation}
where $a\in\mathbb{R}$ and $b>0$ are constants. Now, solve the original
system with the $f^{\prime}(x)^2$ replacing the easier $f^{\prime}(x)$.
However, I can't make sense of substituting the squared term into my
calculations...
Any help or hint is greatly appreciated.