This lesson covers the study of non-linear vibration, focusing on the solution of non-linear differential equations. It begins with a qualitative analysis, followed by an exploration of numerical methods such as the 4th order Runge Kutta method. The lesson explains how to convert second order differential equations into a set of two first order differential equations, and how to use software for solving these equations. It also introduces various functions like ODE 15, ODE 20, ODE 23, and ODE 45. The lesson further delves into approximate methods for finding solutions, such as the straight forward expansion method, harmonic balance methods, Lindstedt Poincare method, and the method of averaging method of multiple scale.

- The Lindstedt Poincare method involves introducing a non dimensional time parameter, Tau, and writing the equation in terms of this Tau.
- The method of harmonic balance involves substituting the function with a Fourier series and equating the coefficients to zero.
- Physical systems can be modeled using non-linear equations and different methods can be used to solve these equations and understand the behavior of the system.

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