Velocity reconstruction from the Orr-Sommerfeld vorticity ($\hat{v}=C_{+}{e}^y+C_{-}{e}^{-y}+{\int }_{y_0}^y\sinh (y-s)W(s)ds$)

The equation $(D^2-1)\hat{v}=W$ is inverted by the displayed formula: differentiating twice produces the endpoint contribution $W(y)$, since $\sinh 0=0$ and $(\sinh {)}^{\prime }(0)=1$. If $W=a_1\mathrm{Ai}(Z)+a_2\mathrm{Bi}(Z)$, 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.