= Solution
Choose $x$ tangential to the solid boundary and locally along the outer inviscid streamlines, and let $y$ be the inward normal coordinate. Write $u$ and $v$ for the corresponding tangential and normal <velocity field> components, $U(x)$ for the outer tangential velocity, and $\nu$ for the <kinematic viscosity>.
For a layer of streamwise scale $L$ and thickness $\delta\ll L$, <incompressible flow> gives
$$
u_x+v_y=0,
$$
so $u=O(U)$ and $v=O(U\delta/L)$. The normal <Navier-Stokes equation> then gives $p_y=0$ to leading order, while the outer <Euler equations for an inviscid fluid>[Euler equations] give
$$
-\frac1\rho p_x=U\frac{dU}{dx}.
$$
In the tangential Navier-Stokes equation, streamwise viscous diffusion is smaller than normal diffusion by $O((\delta/L)^2)$. Retaining the leading inertial and normal-diffusion terms produces the <Prandtl boundary-layer equation>
$$
\boxed{u u_x+v u_y=U\frac{dU}{dx}+\nu u_{yy}}.
$$
It is supplemented by $u=v=0$ at a stationary wall and $u\to U(x)$ on matching to the outer flow.
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