On a class of linear ordinary differential equations having ∑ k = 0 ∞ x k δ ( k ) ( t ) and ∑ k = 0 m x k δ ( k ) ( t ) as solutions

We introduced some linear homogeneous ordinary differential equations which have both formal and finite distributional solutions at the same time, where the finite solution is a partial sum of the formal one. In the nonhomogeneous case and sometimes in the homogeneous case we found formal rational and rational solutions for such differential equations and similarly the rational solution is a partial sum of the formal one.