4 7.36, §4 Solve

Solve the PDE, again effectively a constant coefficient ODE:

$\begin{displaymath} s^2 \hat u = a^2 \hat u_{yy} \end{displaymath}$

$\begin{displaymath} s^2 = a^2 k^2 \quad\Longrightarrow\quad k = \pm s/a \end{displaymath}$

$\begin{displaymath} \hat u = A e^{sy/a} + B e^{-sy/a} \end{displaymath}$

Apply the BC at $y=\infty$ :

$\begin{displaymath} A = 0 \end{displaymath}$

Apply the BC at :

$\begin{displaymath} \hat u_y - p \hat u = \hat f \quad\Longrightarrow\quad - \frac sa B - p B = \hat f \end{displaymath}$

Solving for and plugging it into the expression for $\hat u$ gives:

$\begin{displaymath} \hat u = - \frac{a\hat f}{s+ap} e^{-sy/a} \end{displaymath}$