where α, β and γ are real parameters and thmlsrc">text stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303249&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303249&_rdoc=1&_issn=0022247X&md5=cbdd4368b842b49e933864e14415ac9e" title="Click to view the MathML source">γ>0thContainer hidden">thCode">th altimg="si2.gif" overflow="scroll">γ>0th>. This equation is referred to as Mathieu's equation when thmlsrc">text stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303249&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303249&_rdoc=1&_issn=0022247X&md5=b89fe6caa5ef12ab0463c55b890d7cc3" title="Click to view the MathML source">γ=2thContainer hidden">thCode">th altimg="si3.gif" overflow="scroll">γ=2th>. The parameters determine whether all solutions of this equation are oscillatory or nonoscillatory. Our results provide parametric conditions for oscillation and nonoscillation; there is a feature in which it is very easy to check whether these conditions are satisfied or not. Parametric oscillation and nonoscillation regions are drawn to help understand the obtained results.