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

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Parametric nonoscillation region is given for the Mathieu-type differential equation $$\begin{aligned} x'' + (-\;\alpha + \beta c(t))\,x = 0, \end{aligned}$$ where $$\alpha $$ and $$\beta $$ 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 $$\begin{aligned} y'' + a(t)y' + b(t)y = 0, \end{aligned}$$ where a, $$b: [0,\infty ) \rightarrow \mathbb {R}$$ 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 $$(\alpha ,\beta )$$ and some solution orbits are drawn to help understand the result.