= Solution
Take $k^*>0$ and positive shear scale $S^*$, with kinematic viscosity $\nu^*$. Choose
$$
y=k^*y^*,\quad x=k^*x^*,\quad t=S^*t^*,\quad U=\frac{k^*U^*}{S^*}=y,\quad c=\frac{k^*c^*}{S^*},\quad
\boxed{Re=\frac{S^*}{\nu^*(k^*)^2}.}
$$
The velocity scale is $S^*/k^*$. A reversed shear can be treated by reversing the corresponding coordinate orientation and mode convention. The dimensional <Orr-Sommerfeld equation>, with $U^{*\prime\prime}=0$, is
$$
(U^*-c^*)(D_*^2-(k^*)^2)\widehat v^*
=\frac{\nu^*}{ik^*}(D_*^2-(k^*)^2)^2\widehat v^*.
$$
Substitution of the scales and cancellation gives
$$
\boxed{(y-c)(D^2-1)\widehat v=-\frac{i}{Re}(D^2-1)^2\widehat v.}
$$
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