Properties of solutions of a heterogeneous differential equation of the second order - Mulyava, O.M.; Sheremeta, M.M.; Trukhan, Yu.S. (Carpathian Mathematical Publications, 2019)
Suppose that a power series A(z)=_n=0^∞a_n z^n has the radius of convergence R[A][1,+∞]. For a heterogeneous differential equation z^2 w''+(β_0 z^2+β_1 z) w'+(γ_0 z^2+γ_1 z+γ_2)w=A(z) with complex parameters geometrical properties of its solutions (convexity, starlikeness and close-to-convexity) in the unit disk are investigated. Two cases are considered: if γ_2 eq0 and γ_2=0. We also consider cases when parameters of the equation are real numbers. Also we prove that for a solution f of this equation the radius of convergence R[f] equals to R[A] and the recurrent formulas for the coefficients of the power series of f(z) are found. For entire solutions it is proved that the order of a solution f is not less then the order of A (ρ[f]ρ[A]) and the estimate is sharp. The same inequality holds for generalized orders (ρ_αβ[f]ρ_αβ[A]). For entire solutions of this equation the belonging to convergence classes is studied. Finally, we consider a linear differential equation of the endless order _n=0^∞a_nn!w^(n)=Φ(z), and study a possible growth of its solutions.