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When we work with differential equation modeling of a physical phenomenon, the equation usually has no closed form solution expressible by elementary functions. Engineers and scientists need to understand the behavior of the phenomenon and wish to extract as much information as possible from the equations. In this chapter, we introduce and review several approximation methods by focusing on one of the most important differential equations for which we do not have elementary solution but still we need to understand its behavior with very good approximation. We use the Mathieu equation to be used as a base to introduce some approximate methods that are useful in stability analysis of parametric differential equations. The Mathieu equation is the simplest parametric equation that directly or indirectly appears in stability analysis of dynamic systems.
KeywordsMathieu stability chart Mathieu equation Mathieu functions Parametric vibrations Stability of Mathieu equation Stability chart Recursive method Determinant method Continued fractions Approximate solutions
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