Solution
= Solution
Integrate the axial <ordinary differential equation> twice: $u=-Gy^2/(2\mu)+C_1y+C_2$. The <no-slip boundary condition> at zero gives $C_2=0$, and the one at $h$ gives $C_1=U/h+Gh/(2\mu)$. Hence
$$
\boxed{u(y)=\frac{U}{h}y+\frac{G}{2\mu}y(h-y).}
$$
The first term is <Couette flow>; the second is <Poiseuille flow> driven in positive $x$ when $G>0$. Their addition is valid because the reduced <Stokes equation> is linear.