 Solution

For a boundary layer with exponentially varying outer velocity, the ansatz has $u={\Psi }_Y=e^{(\beta +\gamma )x}{f}^{\prime }(\eta )$. Matching it to $A{e}^{\alpha x}$ requires $\beta +\gamma =\alpha$ and $f^{\prime }(\infty )=A$. 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 $\beta =\gamma$, henceMore explicitly,Canceling the common factor givesTogether with $f(0)=0$, the other two conditions areThey represent wall no slip and outer-flow matching, respectively.