 Solution

For the proposed form,Matching the outer velocity requires $\alpha -\beta =m$. The convective and outer-pressure terms have order $x^{2m-1}$, while $u_{YY}$ has order $x^{\alpha -3\beta }$. Equality of exponents givesThe two terms containing $\eta f^{\prime }{f}^{\prime \prime }$ cancel in $u{u}_x+V{u}_Y$, leaving $x^{2m-1}[m(f^{\prime }{)}^2-\alpha f{f}^{\prime \prime }]$. Since $U_s{U}_s^{\prime }=m{A}^2{x}^{2m-1}$, the reduced equation isThe wall and matching conditions become $\boxed{f(0)=f^{\prime }(0)=0,f^{\prime }(\infty )=A}$. This establishes a consistent similarity reduction; existence of an actual profile is a separate boundary-value question for a particular choice of parameters.