= Solution
Take $\mathbf u=(u(y),0)$ and no imposed streamwise <pressure gradient>. The normal <momentum balance for a material fluid element>[momentum balance] gives $p=p_* -\rho gy\cos\alpha$. The streamwise equation is $\mu u''+\rho g\sin\alpha=0$, since the convective acceleration vanishes. With <no-slip boundary condition>[no slip] at the lower wall and signed upper-wall traction $\tau$ in the positive $x$ direction, the conditions are
$$
u(0)=0,\qquad\mu u'(h)=\tau.
$$
Integration gives
$$
\boxed{u(y)=\frac{\rho g\sin\alpha}{2\mu}y(2h-y)+\frac\tau\mu y,\quad\beta=\frac\rho{2\mu},\quad\gamma=2h+\frac{2\tau}{\rho g\sin\alpha}.}
$$
The required upper-wall <velocity> is its endpoint value $u(h)=\rho gh^2\sin\alpha/(2\mu)+\tau h/\mu$. The unfactored formula remains meaningful when $\sin\alpha=0$.
The fluid exerts shear $\boxed{\mu u'(0)=\rho gh\sin\alpha+\tau}$ per unit length on the lower plane in the positive $x$ direction. The <force> of that plane on the fluid has the opposite sign. The total wall <force> on the fluid balances its downslope weight, providing an independent sign check.
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