= Solution
Normalize $U$ and $c$ by $U_{\max}$ and write $h=\zeta_L\in(0,1)$, $\alpha=kL>0$. In each open region $U''=0$, so the <Rayleigh equation for inviscid shear flow> reduces to $\widehat w''-\alpha^2\widehat w=0$ for the nonreal modes of interest. Incorporating the wall conditions from the start, the <bounded piecewise-linear shear layer> has
$$
\boxed{\begin{aligned}
\widehat w_1&=A\sinh[\alpha(1-\zeta)],\\
\widehat w_2&=B\cosh(\alpha\zeta)+C\sinh(\alpha\zeta),\\
\widehat w_3&=D\sinh[\alpha(1+\zeta)].
\end{aligned}}
$$
These are four independent regional amplitudes before matching. The same expressions extend to noncritical neutral modes and their limits.
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