**1. Introduction**

It is well known that the linear differential equation of the form

$$\sum\_{n=0}^{m} a\_n(t) y^{(n)}(t) = 0, \quad a\_m(t) \neq 0,\tag{1}$$

where *an*(*t*) is an infinitely smooth coefficient for each *n*, and has no distributional solutions other than the classical ones. However, if the leading coefficient *am*(*t*) has a zero, the classical solution of (1) may cease to exist in a neighborhood of that zero. In that case, (1) may have a distributional solution. It was not until 1982 that Wiener [1] proposed necessary and sufficient conditions for the existence of an *N*th-order distributional solution to the differential equation (1). The *N*th-order distributional solution that Wiener proposed is a finite sum of Dirac delta function and its derivatives:

$$y(t) = \sum\_{n=0}^{N} b\_n \delta^{(n)}(t), \quad b\_N \neq 0. \tag{2}$$

It can be easily verified by (10) that *δ*(*t*) is a zero order distributional solution of the equation

$$ty''(t) + 2y'(t) + ty(t) = 0;$$

the Bessel equation

$$t^2y''(t) + ty'(t) + (t^2 - 1)y(t) = 0\natural$$

the confluent hypergeometric equation

$$ty''(t) + (2-t)y'(t) - y(t) = 0;$$

and the second order Cauchy–Euler equation

$$t^2y''(t) + 3ty'(t) + y(t) = 0.5$$

The distributional solutions with higher order of Cauchy–Euler equations were studied by many researchers; see [2–8] for more details.

The infinite order distributional solution of the form

$$y(t) = \sum\_{n=0}^{\infty} b\_n \delta^{(n)}(t) \tag{3}$$

to various differential equations in a normal form with singular coefficients was studied by many researchers [9–13]. Furthermore, a brief introduction to these concepts is presented by Kanwal [14].

In 1984, Cooke and Wiener [15] presented the existence theorems for distributional and analytic solutions of functional differential equations. In 1987, Littlejohn and Kanwal [16] studied the distributional solutions of the hypergeometric differential equation, whose solutions are in the form of (3). In 1990, Wiener and Cooke [17] presented the necessary and sufficient conditions for the simultaneous existence of solutions to linear ordinary differential equations in the forms of rational functions and (2).

As mentioned in abstract, we propose the distributional solutions of the modified spherical Bessel differential equations

$$t^2y''(t) + 2ty'(t) - [t^2 + \nu(\nu + 1)]y(t) = 0$$

and the linear differential equations of the forms

$$t^2y''(t) + 3ty'(t) - (t^2 + \nu^2 - 1)y(t) = 0,$$

where *ν* ∈ N ∪ {0} and *t* ∈ R. The modified spherical Bessel differential equation is just the spherical Bessel equation with a negative separation constant. The spherical Bessel equation occurs when dealing with the Helmholtz equation in spherical coordinates of various problems in physics such as a scattering problem [18].

We use the simple method, consisting of Laplace transforms of right-sided distributions and power series solution, for searching the distributional solutions of these equations. We find that the solutions are in the forms of finite linear combinations of the Dirac delta function and its derivatives depending on the values of *ν*.
