2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230194We obtain explicit solutions for the density $φ_T$ of the first-time $T$ that a one-dimensional Brownian process $B$ reaches the twice, continuously differentiable moving boundary $f$ and such that $f''(t)\geq 0$ for all $t\in \mathbb{R}^+$. We do so by finding the expected value of some functionals of a 3-dimensional Bessel bridge $\tilde{X}$ and exploiting its relationship with first-passage time problems as pointed out by Kardaras (2007). It turns out that this problem is related to Schrödinger's equation with time-dependent linear potential, see Feng (2001).ProbabilityAnalysis of PDEs60J65,45D05,60J60 (Primary); 45G15 (Secondary)text