Put $W=(D^2-1)\widehat v$ and $\delta^3=Re^{-1}$. The constant-shear <Orr-Sommerfeld equation> gives
$$
(y-c)W=-i\delta^3(W''-W),\qquad
W''=\left[1+\frac{i(y-c)}{\delta^3}\right]W
=\frac{i(y-c-i\delta^3)}{\delta^3}W.
$$
Let $a=e^{i\pi/6}$ and $Z=a(y-c-i\delta^3)/\delta$. Since $d/dy=(a/\delta)d/dZ$ and $a^3=i$, the equation becomes
$$
\frac{a^2}{\delta^2}W_{ZZ}=\frac{i}{a\delta^2}ZW,\qquad
\boxed{W_{ZZ}-ZW=0.}
$$
This is the <Airy ordinary differential equation>, yielding the <Airy reduction of the Orr-Sommerfeld equation in constant shear>. It is distinct from the evolution PDE represented by the existing <Airy equation> article.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 3 / d / iii / Solution

{title2=$\widehat v=C_+e^y+C_-e^{-y}+\int_{y_0}^y\sinh(y-s)W(s)\,ds$}

The equation $(D^2-1)\widehat v=W$ is inverted by the displayed formula: differentiating twice produces the endpoint contribution $W(y)$, since $\sinh0=0$ and $(\sinh)'(0)=1$. If $W=a_1\operatorname{Ai}(Z)+a_2\operatorname{Bi}(Z)$, there are four independent constants $C_+,C_-,a_1,a_2$. This reconstructs the velocity from the <Airy reduction of the Orr-Sommerfeld equation in constant shear>; four no-slip and no-flux conditions determine the boundary eigenproblem.

Velocity reconstruction from the Orr-Sommerfeld vorticity

{c}
{title2=$Z=e^{i\pi/6}(y-c-i\delta^3)/\delta,\quad\delta=Re^{-1/3}$}

For nondimensional constant shear $U=y$ and streamwise wavenumber one, put $W=(D^2-1)\widehat v$. The <Orr-Sommerfeld equation> gives $W''=[1+iRe(y-c)]W$. The displayed substitution transforms it into the <Airy ordinary differential equation>, because $(e^{i\pi/6})^3=i$. The scale $Re^{-1/3}$ balances shear advection against viscous diffusion.

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 Incoming links: Airy reduction of the Orr-Sommerfeld equation in constant shear