Write $W(s)=a_1\operatorname{Ai}(Z(s))+a_2\operatorname{Bi}(Z(s))$, using the independent <Airy functions>. To solve $(D^2-1)\widehat v=W$, the <variation of parameters> formula, equivalently a direct convolution, gives the <velocity reconstruction from the Orr-Sommerfeld vorticity>:
$$
\boxed{\widehat v(y)=C_+e^y+C_-e^{-y}
+\int_{y_0}^y\sinh(y-s)\left[a_1\operatorname{Ai}(Z(s))+a_2\operatorname{Bi}(Z(s))\right]ds.}
$$
The reference point $y_0$ is arbitrary. Twice differentiating the integral produces $W(y)$ because $\sinh0=0$ and its derivative at zero is one, verifying the formula. The four constants are independent: applying $D^2-1$ to an identically zero combination first forces $a_1=a_2=0$, then the complementary solutions force $C_+=C_-=0$. The boundary conditions would be $\widehat v(y_j)=D\widehat v(y_j)=0$, $j=1,2$. Their compatibility selects the eigenvalues; they need not be imposed here.

Solution

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

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