Second-order Nonlinear Differential Equations: Oscillation Tests and Applications

Abstract

The 2nd order Differential Equations (DEs) are applied to analyze various phenomena in physics and it is extended to engineering. Specifically 2nd order linear DEs are having a big role in this field. We already observed this working rule in moving systems like a vertical spring attached with a mass. Another example is an electric circuit with a resistor, an inductor, and a capacitor in series connection. These models are applicable to approximate more complex situations like bond structure between atoms or molecules. They are generally patterned as vibrating springs which is explained by the same DEs. Qualitative theory analysis of DEs is an active area of investigation in theory and application point of view. For instance, oscillatory behavior of 2nd order DEs have many applications in the research related to distributed networks where in high-speed computations lossless transmission lines evolve. These lossless transmission lines are employed to connect switching circuits. Recently the study the functional DEs of neutral type is an active area of research. Many researchers are working in the field of oscillation theory and studying only sufficient conditions. Merely few are considering the necessary and sufficient conditions. We have attempted to set up the necessary and sufficient conditions for oscillation of solutions of 2nd order nonlinear neutral DEs. Mathematics Subject Classification (2010): 34K.