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

With $U=U_0$ and $\beta =0$, the Orr-Sommerfeld equation factors into constant-coefficient operators:For distinct nonzero characteristic roots, the general solution isEither square-root choice for $\gamma$ gives the same solution space. The no-slip boundary condition and impermeability impose $\hat{v}=D\hat{v}=0$ at each wall.

The displayed exponential basis needs the usual repeated-root qualification. If ${\gamma }^2={\alpha }^2\ne 0$, replace it by $(a_1+a_{2}y)e^{\alpha y}+(a_3+a_{4}y)e^{-\alpha y}$. 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.