I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times.
The SDE is
$$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$
How does one go about finding an explicit distribution for first passage times in this case?
Your SDE is an example of a linear SDE $$dY_t = (\alpha(t)+\beta(t)Y_t)dt+(\gamma(t)+\delta(t)Y_t)dW_t,$$ where $\alpha$, $\beta$, $\gamma$ and $\delta$ denote deterministic processes and $W$ denotes a one-dimensional Wiener process. In your case, processes $\beta$ and $\delta$ are zero processes and $\gamma$ is a constant process and $\alpha(t) = a+be^{ct}.$ There exists a closed form for the (strong) solution of this SDE, which can be found on Wikipedia. Since your SDE is $$Y_t = A(t) + \sigma W_t,$$ where $A(t) = \int_0^t\alpha(s)ds$ is an exponential curve, studying first passage times of constant levels for $Y$ is equivalent to studying first passage times of exponential curves for the Wiener process.