Nonoscillation of Mathieu’s equation whose coefficient is a finite Fourier series approximating a square wave

Other

Parametric nonoscillation region is given for the Mathieu-type differential equation
x′′+(−α+βc(t))x=0,
where α and β are real parameters. Oscillation problem about a kind of Meissner’s equation is also discussed. The obtained result is proved by using Sturm’s comparison theorem and phase plane analysis of the second-order differential equation
y′′+a(t)y′+b(t)y=0,
where a, b:[0,∞)→R are continuous functions. The feature of the result is the ease of chequing whether the obtained condition is satisfied or not. Parametric nonoscilla- tion region about (α,β) and some solution orbits are drawn to help understand the result.