= Solution
For a <boundary layer with exponentially varying outer velocity>, the ansatz has $u=\Psi_Y=e^{(\beta+\gamma)x}f'(\eta)$. Matching it to $Ae^{\alpha x}$ requires $\beta+\gamma=\alpha$ and $f'(\infty)=A$. The nonlinear terms in the stream-<function> equation scale as $e^{2(\beta+\gamma)x}$, whereas $\Psi_{YYY}$ scales as $e^{(\beta+3\gamma)x}$. Matching exponents gives $\beta=\gamma$, hence
$$
\boxed{\beta=\gamma=\alpha/2.}
$$
More explicitly,
$$
\Psi_Y\Psi_{xY}-\Psi_x\Psi_{YY}
=e^{2\alpha x}\left[\alpha(f')^2-\frac\alpha2ff''\right],\qquad
\Psi_{YYY}=e^{2\alpha x}f''',\qquad U_sU_s'=\alpha A^2e^{2\alpha x}.
$$
Canceling the common factor gives
$$
\boxed{f'''+\frac\alpha2ff''+\alpha\bigl(A^2-(f')^2\bigr)=0.}
$$
Together with $f(0)=0$, the other two conditions are
$$
\boxed{f'(0)=0,\qquad f'(\eta)\longrightarrow A\quad(\eta\to\infty).}
$$
They represent wall no slip and outer-flow matching, respectively.
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