= Solution
{c}
At zero surface <shear stress>, the solution connected to part (a) is
$$
\boxed{f_0(\zeta)=\zeta.}
$$
Indeed, it has $f_0'=1$, $f_0''=f_0'''=0$ and $f_0(0)=0$. Insert the <regular perturbation> $f=\zeta+\gamma f_1+O(\gamma^2)$ into the nonlinear <ordinary differential equation>. The terms are
$$
f'''=\gamma f_1'''+O(\gamma^2),\qquad
ff''=\gamma\zeta f_1''+O(\gamma^2),\qquad
1-(f')^2=-2\gamma f_1'+O(\gamma^2).
$$
The coefficient of $\gamma$ therefore gives
$$
\boxed{f_1'''+\zeta f_1''-2f_1'=0.}
$$
Expanding the surface <boundary conditions> and the far-field matching condition gives
$$
\boxed{f_1(0)=0,\qquad f_1''(0)=1,\qquad
f_1'(\zeta)\longrightarrow0\quad(\zeta\to-\infty).}
$$
\b[The printed $+\infty$ limit is a sign error.] It is present in the native PDF on paper page 3, not introduced by transcription. The fluid occupies $z<0$, the <inner variable> is $\zeta=z/\delta<0$, and part (b)(iii) correctly matches at $-\infty$. We retain that physical domain throughout. A change to the positive coordinate $\eta=-\zeta$ would also change the differential equation and surface <derivative> signs; it cannot be made by changing only the far-field sign.
The inequality $\delta^2\ll\gamma$ ensures that effects discarded at order $\delta^2$ in the <boundary-layer scaling> are smaller than the retained order-$\gamma$ perturbation. The inequality $\gamma\ll1$ justifies this <regular perturbation>. The two conditions control different approximations.
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