A generalization of the damped pendulum equation discussed in the text, or a damped

Chapter 9, Problem 27

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A generalization of the damped pendulum equation discussed in the text, or a damped spring-mass system, is the Linard4 equation d2x dt2 + c( x) dx dt + g( x) = 0. If c( x) is a constant and g( x) = kx, then this equation has the form of the linear pendulum equation (compare with equation (12) of Section 9.2 with sin replaced by its linear approximation, ); otherwise, the damping force c( x) dx/dt and the restoring force g( x) are nonlinear. Assume that c is continuously differentiable, g is twice continuously differentiable, and g(0) = 0. a. Write the Linard equation as a system of two first-order equations by introducing the variable y = dx/dt. b. Show that (0, 0) is a critical point and that the system is locally linear in the neighborhood of (0, 0). c. Show that if c(0) > 0 and g_(0) > 0, then the critical point is asymptotically stable, and that if c(0) < 0 or g_(0) < 0, then the critical point is unstable. Hint: Use Taylor series to approximate c and g in the neighborhood of x = 0.