Nonclassical behavior of solutions of linear second-order ordinary differential equations (Q944437)

The author presents unstable second-order equations of the form \[ \ddot y + (1 + g(x))y = 0, \] where \(g(x)\in C(0,\infty )\) and \(\lim_{x\to\infty} g(x) = 0,\) but the maximum absolute values of solutions grow unboundedly (as power-law functions or even as exponentials). Two examples are given as follows: (1) For the equation \(y'' + (1 +\frac{8 \sin 2x} {2x+\sin 2x})y = 0,\) the unbounded solution is \(y =(2x + \sin 2x) \cos x\). (2) For the equation \(y'' + (1 -\frac{8 \sin 2x} {2x-\sin 2x} )y = 0\), the unbounded solution is \(y =(2x - \sin 2x) \sin x\).