Abstract
In this chapter we introduce a new variation of parameters formula and apply the idea to ordinary and delay differential equations with constant or variable delays. It is customary in a nonlinear differential equation to add and subtract a convenient term that allows us to invert the equation in question and obtain a variation of parameters formula, which can be used to obtain different results on the solutions. However, the added term will cause restrictions on the coefficients and, as a result, limit the class of equations that can be considered. Our method is to introduce a new function and use it to obtain the integrating factor and then invert to get an integral relation, which we call a new variation of parameters formula. Once that is done, we obtain results concerning the existence of periodic solutions and stability of the zero solution. The results of this chapter are totally new and should serve as foundations for future research.
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