$6\mathrm{.}3.$ In the laminar boundary layer on a flat plate, the velocity profile is assumed to be represented by:

$\frac{U}{U_{}}=a_0+\frac{a_1}{b}ar(y)+\frac{a_2}{b}ar(y{)}^2+\frac{a_3}{b}ar(y{)}^3\frac{,}{b}ar(y)=\frac{y}{}$

If $a_0($etc$.)$ are constants, show that in order to satisfy the appropriate boundary conditions for a zero pressure gradient flow:

$a_0=0,a_1=\frac{3}{2},a_2=0,a_3=-\frac{1}{2}$

Use the Momentum Integral Equation $($MIE$)$ to derive the law governing the rate of growth of boundary layer thickness, $\frac{}{x}=4\mathrm{.}64R{e}_x^{-\frac{1}{2}},$ and the skin friction law for the boundary layer, $C_f=0\mathrm{.}646R{e}_x^{-\frac{1}{2}}.($Tip: start from boundary conditions for the wall and the edge of the boundary layer$)$