Example 1.  Solve the second order  I.V.P.  
      with  ,   .    
Use the Runge-Kutta method to compute the solution over the interval .  

Solution 1.

The D. E. can be written as     where  .  
We use the substitution   and make the second order I.V.P. into a system of first order I.V.P.'s  

    ,  
and  
    ,  
where  
    ,  
and  
     ,   .  

Compute the Runge-Kutta solution.

The solution we seek is the first coordinate in the 2D system.

Now we can plot the solution.

(c) John H. Mathews 2004