Example 7.   Solve  ,  by using the Runge-Kutta method to solve the system of D. E.'s  
         ,  
and
        .    

Solution 7.

Enter the functions    and    and form the vector function  .  

Compute the Runge-Kutta solutions.
First, graph four spirals around the origin.

Combine the above graphs.

Second, graph a curves where the initial velocity is large enough to make the pendulum turn twice about its pivot.

Third, graph a curves where the initial velocity is large enough to make the pendulum turn three about its pivot.

Combine the above graphs.

Third, graph two curves where the initial velocity is large enough to make the pendulum continue turning about its pivot but in the opposite direction.
Remark.  In order to keep these curves from plotting in quadrant III, we have started them at instead of .

Combine the above graphs.

Next, graph four spirals around the point .

Combine the above graphs.

Next, graph two curves where the initial velocity is large enough to make the pendulum continue turning about its pivot but in the opposite direction.
Remark.  In order to keep these curves from plotting in quadrant III, we have started them at instead of , and the time interval is larger than before.

Combine the above graphs.

(c) John H. Mathews 2004