A pendulum with a rod of length 32 feet is deformed pi/6 radians from the vertical position and released (initial velocity 0 at time t=0).  The governing differential equation (neglecting damping) is given below (See page 249 in Zill or this development of the equation.).  The bottom picture at the right shows the initial position of the pendulum (just as it is released at time t=0).  Click on the picture to see an animation.  Click here to see an animation represented in a scaled coordinate system.  The animation corresponds to the solution to the linear model.

Here are two wonderful applets.  Simple pendulum is similar to my animation but allows you to interactively change the initial condition, follows the nonlinear model, and allows you to compare the fundamental periods for the linear and nonlinear models.  It is really quite nice.  Spring pendulum uses a spring for the rod.  There does not exist a differential equation able to model this motion.  Take a look.

The graph in red is of y = x and the graph in blue is of y = sin(x).  The vertical graphs in green are of x = -pi/6 and x = pi/6.  Notice that x is a fairly good approximation for sin(x) between the vertical green lines.

Click here to see the Maple code for generating terms in the series solution and graphing truncated series solutions along with the analytical solution to the linear model.  x is being used for t in the Maple code.  Many coefficients are zero.  

If the initial conditions had instead been

then the linear model would be a less accurate approximation.  Click here to see a Maple worksheet using these initial conditions.

Here is a nonlinear oscillating pendulum applet that follows a different nonlinear model (it includes dampening) and allows you to swing the pendulum right over the top.  Maple Worksheet

It must be noted in these last models that since the equations are not linear, our textbook theory regarding convergence of power series solutions cannot be applied.  It is still fun to investigate solutions when we have a powerful tool like Maple at our disposal.  It appears that the power series solutions have finite intervals of convergence.