= Solution
With $U=U_0$ and $\beta=0$, the <Orr-Sommerfeld equation> factors into constant-coefficient operators:
$$
(D^2-\alpha^2)(D^2-\gamma^2)\widehat v=0,\qquad
\boxed{\gamma^2=\alpha^2+iRe(\alpha U_0-\omega).}
$$
For distinct nonzero characteristic roots, the general solution is
$$
\boxed{\widehat v=a_1e^{\alpha y}+a_2e^{-\alpha y}+a_3e^{\gamma y}+a_4e^{-\gamma y}.}
$$
Either square-root choice for $\gamma$ gives the same solution space. The <no-slip boundary condition> and impermeability impose $\widehat v=D\widehat v=0$ at each wall.
The displayed exponential basis needs the usual repeated-root qualification. If $\gamma^2=\alpha^2\ne0$, replace it by $(a_1+a_2y)e^{\alpha y}+(a_3+a_4y)e^{-\alpha y}$. If just one root pair is zero, that pair contributes $1,y$; if both pairs are zero, the basis is $1,y,y^2,y^3$. These are the complete limiting cases, not four independent copies of a repeated exponential.
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