Referring to $X(t)$ and $\tau(x) = \inf \{ t>0 : X(t)<0 \mid X(0) = x \}$ stated above, a Feynman-Kac formula implies that the function $u(x) = \mathbb{E}^x \exp(-\tau)$ satisfies a second-order, linear differential equation: $$ \begin{cases} \frac{1}{2} \sigma(x)^2 u''(x) + u'(x) - u(x) = 0 \\ u(0)=1\;, \quad u(\infty) = 0 \end{cases} $$ If $\sigma(x)^{-2}$ is integrable, then these equations can be put in the form of a classical Sturm-Liouville problem on a semi-infinite interval. For properties of their solutions see, e.g., Zettl, Anton (2005). Sturm–Liouville Theory. Providence: American Mathematical Society.

Referring to $X(t)$ and $\tau(x) = \inf \{ t>0 : X(t)<0 \mid X(0) = x \}$ stated above, a Feynman-Kac formula implies that the function $u(x) = \mathbb{E}^x \exp(-\tau)$ satisfies a second-order, linear differential equation: $$ \begin{cases} \frac{1}{2} \sigma(x)^2 u''(x) + u'(x) - u(x) = 0 \\ u(0)=1\;, \quad u(\infty) = 0 \end{cases} $$ If $\sigma(x)^{-2}$ is integrable, then these equations can be put in the form of a classical Sturm-Liouville problem on a semi-infinite interval. For properties of their solutions see, e.g., Part 4 of Zettl, Anton (2005). Sturm–Liouville Theory. Providence: American Mathematical Society.