For each nonlinear equation of motion, linearize the equation about the indicated equilibrium position and discuss the range of validity of the linearized equation. Plot the relative error for the linearized equation of motion. Analytically solve the linearized equation of motion.

(a) \(\ddot{\theta}+3 \cos \theta=0, \theta_{e q}=-\pi / 2\), and zero initial conditions.

(b) \(\ddot{\theta}+3 \sin \theta=0, \theta_{e q}=0, \theta(0)=0.5\), and \(\dot{\theta}(0)=0\).

(c) \(\ddot{\theta}+3 \cos ^{2} \theta=0, \theta_{e q}=-\pi / 2\), and zero initial conditions.

(d) \(\ddot{\theta}+3 \theta^{3}=3, \theta_{e q}=1\), and zero initial conditions.

(e) \(\ddot{\theta}+\sqrt{\theta}=1, \theta_{e q}=1\), and zero initial conditions.