This paper revisits the use of the Laplace transform in the study of Bessel and modified Bessel differential equations. Rather than presenting the method as a new alternative to the classical Frobenius approach, the paper is organized as a pedagogical review of how standard operational rules for the Laplace transform lead to transform-domain differential equations for the regular solutions of these variable-coefficient problems. For Bessel’s equation, the transformed equation yields the classical Laplace transform of Jν(x); for the modified Bessel equation, the same procedure yields the corresponding transform of Iν(x). These formulas are then used to recover standard recurrence and derivative identities and to derive several illustrative transform evaluations. The paper also explains why the singular companion solutions Yν(x) and Kν(x) are not obtained directly from the regular Laplace-transform framework and must instead be introduced through classical connection formulas. Particular attention is given to placing the calculations in the context of the earlier literature, especially the classical treatise of Watson and later work on Laplace and Lipschitz–Hankel integrals. In this form, the paper is intended as a self-contained review and tutorial on a useful operational approach to Bessel-type equations.
Osman Yürekli (Thu,) studied this question.