= Solution
The backward equations for the two mean hitting times are
$$
\begin{aligned}
-1&=s(\tau^+)' +\lambda(\tau^--\tau^+),\\
-1&=-s(\tau^-)' +\lambda(\tau^+-\tau^-),
\end{aligned}
$$
with $\tau^-(0)=0$ and $\tau^+(L)=\tau^-(L)$. Their difference obeys
$$
\frac d{dy}(\tau^+-\tau^-)=-\frac2s,
$$
and the reflecting condition fixes $\tau^+-\tau^-=2(L-y)/s$. Integration then gives
$$
\boxed{\tau^-(y)=\frac ys
+\frac{2\lambda}{s^2}\left(Ly-\frac{y^2}{2}\right),}
$$
and
$$
\boxed{\tau^+(y)=\frac{2L-y}{s}
+\frac{2\lambda}{s^2}\left(Ly-\frac{y^2}{2}\right).}
$$
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