Finite series of distributional solutions for certain linear differential equations

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations t²y^′′ (t) + 2ty^′ (t) − [t² + ν(ν + 1)]y(t) = 0 and the linear differential equations of the forms t²y^′′ (t) + 3ty^′ (t) − (t² + ν² − 1)y(t) = 0, where ν ∈ N ∪ {0} and t ∈ R. We find that the distributional solutions, in the form of a finite series of the Dirac delta function and its derivatives, depend on the values of ν. The results of several examples are also presented.