Linear ordinary differential equations of the second order with unbounded solutions

摘要

We consider three families of equations of the form y″ + (1 + φ(x))y = 0, where the coefficient φ(x) satisfies the condition lim x→+∞ φ(x) = 0. We obtain solutions of these equations in closed form. We show that the maximum absolute values of solutions grow at the rate of a logarithmic function, a power-law function, and even an exponential function as x → ∞.