= Solution
{c}
The two external stream <velocities> impose
$$
\boxed{F'(+\infty)=\frac{\Delta U}{2U_c},\qquad
F'(-\infty)=-\frac{\Delta U}{2U_c}.}
$$
Let $r=\Delta U/U_c\ll1$ and seek $F=O(r)$ over $\xi=O(1)$. The nonlinear term $FF''$ is then $O(r^2)$, so the leading <ordinary differential equation> is
$$
F'''+\xi F''=0.
$$
For $G=F''$, this is the first-order <linear differential equation> $G'+\xi G=0$. Its <integrating factor> is $e^{\xi^2/2}$, giving $F''=M e^{-\xi^2/2}$. An <integration> using the <error function> gives
$$
F'=M\sqrt{\frac\pi2}\operatorname{erf}(\xi/\sqrt2)+E.
$$
The two far-field <boundary conditions> require $E=0$ and $M=r/\sqrt{2\pi}$. A second <integration> uses
$$
\int\operatorname{erf}(\xi/\sqrt2)\,d\xi
=\xi\operatorname{erf}(\xi/\sqrt2)+\sqrt{\frac2\pi}e^{-\xi^2/2}.
$$
Consequently
$$
\boxed{F=A\xi\operatorname{erf}(\xi/\sqrt2)+Be^{-\xi^2/2}+D,
\qquad A=\frac{\Delta U}{2U_c},\quad
B=A\sqrt{\frac2\pi}=\frac{\Delta U}{U_c\sqrt{2\pi}}.}
$$
The relation between $A$ and $B$ cancels the extra $\xi e^{-\xi^2/2}$ term when differentiating. It gives the familiar <error function> mean <velocity> profile
$$
U=U_c+\frac{\Delta U}{2}\operatorname{erf}(\xi/\sqrt2)
$$
to first order in $r$. The additive constant $D=O(r)$ does not affect this <velocity> profile at the retained order; its interpretation comes from the transverse <velocity>.
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