Solution
= Solution
The wall speed sets
$$
\boxed{U(x)=\alpha x}.
$$
Using $x$ as the local streamwise length gives
$$
\operatorname{Re}_x=\frac{U(x)x}{\nu}
=\frac{\alpha x^2}{\nu},
$$
which is large when $x\gg\sqrt{\nu/\alpha}$. The <boundary-layer scaling>
$$
\frac{U^2}{x}\sim\frac{\nu U}{\delta^2}
$$
then gives
$$
\delta(x)\sim\sqrt{\frac{\nu x}{U(x)}}
=\boxed{\sqrt{\frac{\nu}{\alpha}}},
$$
so the stretching-sheet layer has constant thickness to leading order.