Nonclassical behavior of solutions of linear second-order ordinary differential equations

We present a number of unstable second-order equations of the form

$$y'' + (1 + g(x))y = 0,$$

where the coefficient g(x) satisfies the conditions g(x) ∈ C(0, ∞) and lim_x→+∞ g(x)=0 but the maximum absolute values of solutions grow unboundedly (as power-law functions or even as exponentials).