= Solution
Write $R_e=R^+$ and apply the stated constant-coefficient <boundary value problem>. A constant particular solution is $-S/(DB^2)$. Regularity at the origin removes $\cos(Br)/r$, so
$$
\phi(r)=A\frac{\sin(Br)}r-\frac{S}{DB^2}.
$$
The outer <boundary condition> gives $A=SR_e/[DB^2\sin(BR_e)]$, hence
$$
\boxed{\phi(r)=\frac{S}{DB^2}\left[\frac{R_e\sin(Br)}{r\sin(BR_e)}-1\right].}
$$
At $r=0$, use the <limit> $\sin(Br)/r\to B$ rather than substitute into the apparent $0/0$. The result satisfies both the differential equation and $\phi(R_e)=0$ provided $\sin(BR_e)\ne0$. This is the ideal extrapolated-boundary model specified in the question; a separately resolved outer transport layer would need its own matching conditions.
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