Construction of algebraic-differential equations with given smooth and impulsive behaviour

The paper is consisted of two parts. In the first part, we are presenting the smooth and impulsive solution space B of the system of algebraic and differential equations A (ρ)β (t) = 0, ρ := dt where A (ρ) ∈ R [ρ] (with detA (ρ) 6= 0) is given. The solution space is given in terms of the finite and infinite Jordan pairs of the polynomial matrix A (ρ), and therefore a necessary introduction is given for the finite and infinite zero structure of the polynomial matrix A (ρ). The second part, studies the inverse problem : Given a smooth and impulsive solution space B, try to find out a system of algebraic and differential equations A (ρ)β (t) = 0, ρ := dt with the given solution space.