= Solution
Use <autonomous third-order reduction using the dependent variable>. On a monotone branch where $f$ can serve as the independent variable, the <chain rule> gives
$$
f''=GG_f,\qquad f'''=G(G_f^2+GG_{ff}).
$$
Therefore the reduced second-order equation is
$$
\boxed{G^2G_{ff}+G(G_f)^2+\frac\alpha2 fGG_f+\alpha(A^2-G^2)=0.}
$$
The wall and matching conditions become
$$
\boxed{G(0)=0,\qquad G(f)\longrightarrow A\text{ as }f\to\operatorname{sgn}(A)\infty.}
$$
For the usual positive outer flow this is the limit $f\to+\infty$. The reduced equation is singular at $G=0$: if $f''(0)=b\ne0$, then $f\sim b\eta^2/2$ and $G\sim b\eta$, so $G$ has square-root behavior as a <function> of $f$ near the wall. A finite value for $G_f(0)$ should not be imposed as an extra <boundary condition>.
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