= Solution
Expand $\psi=\epsilon\psi_1+\epsilon^2\psi_2+\cdots$ and $U=\epsilon U_1+\epsilon^2U_2+\cdots$. A <Taylor expansion> of the sheet boundary conditions about $y=0$ gives
$$
\nabla^4\psi_1=0,\qquad \psi_{1y}(0)=0,\qquad \psi_{1x}(0)=\cos\theta.
$$
The first-order <Fourier mode> is $\psi_1=f(y)\sin\theta$, with $(\partial_y^2-1)^2f=0$, $f(0)=1$, and $f'(0)=0$. Decay excludes the growing exponentials, leaving $f=(a+by)e^{-y}$. The two boundary values give $a=b=1$:
$$
\boxed{\psi_1=(1+y)e^{-y}\sin\theta,\qquad u_1=-ye^{-y}\sin\theta,\qquad v_1=-(1+y)e^{-y}\cos\theta.}
$$
There is no first-order mean boundary motion. Boundedness excludes mean shear, so $\boxed{U_1=0}$.
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