= Solution
Starting from $+1$, the level reaches $c$ at time $c$ if the first holding time exceeds $c$. This has probability $e^{-ac}$; surviving beyond $c$ also makes the level strictly greater than $c$, as required by the strict event. A switch to $-1$ at time $u<c$ has density $qe^{-au}\,du$, leaves level $u$, and requires a later rise of $c-u$. Absorption before reaching $c$ cannot succeed. Therefore
$$
\boxed{\psi_+(c)=e^{-(q+\lambda)c}+\int_0^cqe^{-(q+\lambda)u}\psi_-(c-u)\,du.}
$$
Holding times have continuous distributions, so the single endpoint $u=c$ has no effect on the formula.
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