Two different equations of motion from the same Lagrangian?

[Happiness]

The equation of motion of a pendulum with a support oscillating horizontally sinusoidally with angular frequency $\omega$ is given by (5.116). (See attached.)

I get a different answer by considering the Euler-Lagrange equation in $x$ and then eliminating $\ddot{x}$ in (5.115):

Referring to (5.114), we have
$\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=\frac{\partial L}{\partial x}$
$\frac{d}{dt}(m\dot{x}+ml\dot{\theta}\cos\theta)=0$
$m\ddot{x}+ml\ddot{\theta}\cos\theta-ml\dot{\theta}^2\sin\theta=0$
$\ddot{x}=l\dot{\theta}^2\sin\theta-l\ddot{\theta}\cos\theta$

Substituting this into (5.115), we have
$l\ddot{\theta}+l\dot{\theta}^2\sin\theta\cos\theta-l\ddot{\theta}\cos^2\theta=-g\sin\theta$
$l\ddot{\theta}\sin\theta+\dot{\theta}^2\cos\theta=-g$

The solution $\theta (t)$ would in general be different from the solution $\theta (t)$ of (5.116). Why are there two solutions? What does the former solution represent?

 

[composyte]

The reason you're getting a different answer is because you are not solving the same system. The problem gives you x (t), but you are trying to solve for it as though it is an unknown. By doing this you are changing the support from moving at a known oscillation to having its oscillating based on the swinging of the pendulum. Hope that helps